Hi I would say the problem is the chipboard,he should never have used this,what he should have used was a wpb plywood of a thickness of at least 12mm then overboarded with the same again and used schluter ditra uncoupling matt.The trouble with chipboard is it take,s in any moisture and with underfloor heating there will be a lot of movement that,s why the tiles are loose In my opinion there is no cure apart from taking it all up and starting afresh sorry.

Your problems could be caused due to many different reason. You would never be 100% accurate on the cause unless you carefully remove the existing floor tiles and investigate. The tiles if removed carefully could be used again. The problems could be ill placed joist / flooring boards. Or this could all be find and the past builder might of laid the tiles using a small amount of adhesive on each corner of the tile and now they are in use they are moving. This is caused 'buttering the corners' and really isnt a method any one should use. Lastly, when laying onto of wooden flooring you must use either a flexible adhesive or if using a normal adhesive use a flexible add mixture to it. If not when the wood expands ect as wood does the adhesive will crack. If it is investigated by a competent firm such as ours, your kitchen should go untouched.

You can't say the kitchen will go untouched without knowing what's tiled. What if the floor was tiled before the kitchen went in? I'm guessing it did as the units have moved. A minimum of 15mm ply must be used as a wooden substrate when tiling, this is a british standard requirement. Most adhesives recommend a minimum of 18mm though so read the product your usings guidelines before laying anything. The whole floor has to be removed. The floor tiles might be able to be reused but the time you would be paying a tiler "if you would find one to do it" to remove adhesive ect from the tiles you may as well get new ones. Sorry to be the barer of bad news but his is the case. Flexible adhesive must be used but it must be suitable for the porcelain your using. If there unsealed white/light polished porcelain you must use a white adhesive, seal them, then grout. If your using a black/dark polished porcelain you must use dark adhesive, seal them, then grout. This is because they are pouruss and you will see patches in the tiles if this is not done.

I'm afraid I can only give my opinion regarding possible problems with the tiles. If there is a problem with the joists ect I really do feel sorry for you as its going back to the start to solve it.

Recently installed wet pipe underfloor heating in an extension. The floor joist centers were 400mm but still put on 25mm Marine Plywood (we were concerned by weight of kitchen units filled with heavy crockery etc as well as thick granite woektops, so we went overboard or so we thought) , then Detra Mat, Flexible adhesive and grout.
Still have slight flex in the floor when heavy person moves around in the kitchen. Nothing dramatic as there are no cracks in the floor, but there is noticeable movement in the floor and kitchen units.
Back to yours- Chipboard is not good! You might get away with leaving the units in place as the underfloor heating may not have been run under.
I would not attempt just a tiling repair, without first getting to the floor joists in one area. Check size of joists, centers and span, check with engineer that these are OK. Once Ok replace the chipboard with marine ply.