Ttdfjt

You should work on the physics behind the floating rocks. What is the principle that makes them float?

We know that the force of the stones is parallel (but opposite in verse) to the gravity, so it alway points toward the sky.

This means that the rocks can only make the castle stay afloat. In order to move the castle, you need a force that can be addressed, with a parallel component with respect to the surface.

You said that, far from the Line, the floating power of the rocks gradually vanishes, as well as it is weaker the higher you are from the surface, so we can make some assumptions.
Everything sounds as if there is a repulsive force from the ground (1), as well as a material that can dampen the force toward the sky that is generated by such stones (which should be plentiful far from the Line).

Now, let's say that I can shield my floating stones:

^{The mortals tremble in the presence of my superior Paint skills!}

An unshielded stone, receives repulsive force from all directions, so that all the components parallel to surface cancel themselves. A shielded one, receive only the forces from one direction, so that it has a smaller component toward the sky, but also a component parallel to the surface, pointing in the opposite verse with respect to the opening in the shielding.

So, a castle could have some big unshielded stones, mainly used to keep everything afloat, and one or more partially shielded stones, mounted on joints that can be orientated, to propel and steer the castle itself.

(1) But let's be careful: the repulsive force from something that can be approximated as an infinite plane would be the same at every distance from the plane itself (in other words, it wouldn't fade with height). I suppose that the Line is thin enough that the ground under the castle can't be considered an infinite, repulsive, plane.

Brute force. Lots of inertia in a castle, so you'll have problems making an engine sufficiently powerful to drive a propellor to move it along, even if you had substantially more advantaged technology than was generally available in the medieval period.

Therefore, brute force.

One means of brute force is to just get a lot of slaves on the end of ropes. This might be a good reason to get back to the good old loot'n'enslave warfare that was so popular back in the axial age. A slightly less ethically challenged option would be to use lots of draft animals... oxen would probably do, because you don't want to get up too much speed with a castle.

Another possibility is to use anchors and capstans, and use mechanical advantage to haul your castle up towards an anchor, and then (slightly less) brute force to move your anchor points in the direction you wish to go. You might have to use this method to make headway against very strong winds, and you'll definitely want to be anchoring your castle in place when you're not moving it around.

There's a minor additional benefit here in that you could perhaps fire a lightweight anchor via a siege weapon, and therefore be able to haul yourself along without actually putting troops or animals on the ground. This would be useful for navigating hostile or otherwise impassible terrain such as mountains, swamps or shallow lakes and rivers.

Ley-Line Canals

https://commons.wikimedia.org/wiki/File:Segovia_Aqueduct.JPG

Your castles may float along dedicated canals which are specifically built to allow for floating castles to navigate them. The Romans famously built aqueducts for transporting water, if you built something similar, it would allow for you to easily transport your castles around.

The canals would be constructed along the paths of the ley-lines, essentially acting as guides to prevent the castle from drifting away from the ley-line. This also means you will not accidentally drift into a “dead zone” where the stones do not float at all, which would cause a castle to crash.

Sails

Canals alone though wont cause you to move forward. Instead, as you are now floating along fixed lines, like a railroad track, you can now safely use sails to push you forwards. Before, without he canal, you risked being pushed out of the ley-line. Now though, as you are more firmly fixed into the line, there is significantly less of a chance for you to be pushed out, most momentum from the wind should push you forwards (assuming you angle your sails correctly and the wind is not blowing in the complete opposite direction you want to go).

Wheels

Alternatively, you might opt to use wheels to move your castles. Wheels would run along grooves in the canal and could be powered through a number of ways, such as by man power, using beasts of burden or even with falling water like a watermill. Don’t expect the castle to move particularly quickly though.

One problem with the concept of using canals though is they would take a very long time to build everywhere. Perhaps you only have canals on major routs, such as between cities (much like how highways only run between large cities and smaller towns build up along those routs). You could also take from the idea of shipping lanes where canals are only built on well-established and often used routs.

Magnetism

Essentially, your castles could work like levitating trains. Your stones would work like magnets, repelling away from the ley-line which causes them to float. If you can invert that signal, causing them to be pulled towards the ley-line rather than being pushed away, you could propel yourself through the power of magnetism (not really magnets per say but they function similarly enough).

The way magnetic trains work is by turning on and off magnets in a certain order, causing the train to be pulled forward. Your stones could be turned on and off, when they are on, they float on the ley-line, when they are off, they fall.

So how can we turn this into forward movement? Well if we start with all of the stones being turned on, the castle will be floating and relatively stationary (barring the flow of the ley-line and the wind pushing it). What we do next is turn off the stones at the front of the castle, only for a fraction of a second, which will cause the castle to tilt slightly forwards. The force of gravity will start pulling the castle downwards on that side, pulling it forward. To prevent us from crashing, we turn the stone back on, floating us back up, only now we are slightly further forward. If we keep repeating this process rapidly, we will soon start to build up significant speed, despite the castles high mass.

This concept is very similar to that of the rail gun which uses electromagnetic force to move a metallic object at high speed. Contrary to its name, it is not just intended for weaponry, NASA proposed to use a rail gun to accelerate small aircraft or spacecraft into orbit..

Tracks.

I wondered for how long people had been using rail-type tracks to move cargo overland. A long time: I found the Diolkos, used by the ancient Greeks to move ships overland.

https://en.wikipedia.org/wiki/Diolkos

The Diolkos was a trackway paved with hard limestone[26] with parallel
grooves running about 1.60 metres (63 in) apart.[31] The roadway was
3.4 to 6 metres (11 to 20 ft) wide.[26] Since ancient sources tell little about how the ships were hauled across,[24] the mode of ship
transport has largely to be reconstructed from the archaeological
evidence. The tracks indicate that transport on the Diolkos was done
with some sort of wheeled vehicle.

Your ley lines would have parallel tracks along them, probably of stone. Stays down within the tracks would prevent upward motion of the castle and having two would prevent lateral movement out of the influence of the ley lines supporting power.

The castle would be moved by sliding the stays along the tracks. A team of oxen could do this, in the same manner an ox team might pull a barge. Depending on the distance between the two stays you might need 2 teams working together.

You could do away with the stays and rails and just use the ox teams but you would need to have a cart heavy enough to oppose the lifting power of the castle which is more work to move.

What instantly comes to mind is: WATER

Water can be collected from rainfall in a cistern within the castle.
You could then easily direct the flow of the captured water, and with water being relatively dense it would have the kind of propulsion you're looking for.

Since there is mechanical power available to these castles as the question stated,
you could also manipulate the pressure at which water is propelled. Think of it like an ancient fire hose, which can output water with an immense amount of force.

They have lifting stone in 'The Edge Chronicles' and ships all employ a Stone Pilot to carefully control the temperature of the ship's stone. Hot stone rises and cold stone falls, but differential heating can be used to angle the lift vector and push the ship along.

High performance stones have carefully bored holes, to allow fire (or cooling air) to reach the middle of the stone. Most people understand the concept, but only the stone pilots have the finesse to safely control a ship.

Take a look at the work of Leonardo Da Vinci, he covered a couple of designs related to flight such as a man powered machine with mechanical wings, and a aerial screw. Either of these could potentially by modified and increased in power and size to assist in moving a floating castle. They would obviously need to be massively increased in size and efficiency to move something so massive though. Horse powered treadmills could provide the mechanical energy. As you said, they've had a few decades to improve on initial solutions so perhaps they may have increased to the point of developing a reasonable propeller.
Remember too that the ancient Greeks developed the first steam engine, but saw it as a toy rather than as something that could be used as a tool.

You should be able to sail the castle, the issue would be sail material, you'd be using bamboo, timber and metal rather than cloth because the structures are going to be too large for traditional fabric. Controlling sails that large will be challenging but not impossible, I'd think that capstan and treadwheel driven winches would be sufficient.

Fun fact about late Roman technology: They had all the tools to make a steam engine.

http://www.foresightguide.com/50CE-a-steam-engine-in-ancient-rome/

Steam engines are useful for a lot of things, and would help with several of the previous answers. Whether you're going on water or land having a steam engine would help enormously.

That may be a bit steampunk, and it assumes that the knowledge wasn't lost (they were called the dark ages for a reason), but steam turbines and propellers could definitely move your castle along the line.