We bought our house new from a local builder and have had problems with the kitchen floor since day 1. It is a large room (6.45 x 4.25m) with 60x60cm Porcelanosa tiles. These tiles started clicking and the grouting cracking so we complained to the builder. Its a long story, but they initially investigated and the 6 tiles they investigated popped up. They then agreed to replace the tiles, but when it came to it, just replaced the tiles that were proving an issue. Eventually we got them to agree to re-lay all tiles in February. Immediately we had movement, with the grouting cracking badly, tiles lifting and moving under weight. After suggesting many fixes, I did some research on why it might be happening and came up with deflection. I then used a glass of water to test the movement and discovered that in one area in particular, the floor (minus tiles) was moving far too much. The floor is concrete base, celotex, chipboard and ply, with flexi adhesive and grout. There was some kind of membrane also but I'm not clear on this or the dimensions of each. Today they removed a section of the floor (in the affected area) and found no evidence of damp or any other problem. Their proposed solution is to take up the family room end of the room (the other end has all the kitchen units, granite worktops and a huge granite island unit) and renew all the layers. We're not sure this is going to solve the problem - any experience here hugely appreciated. We're contemplating getting a surveyor in and paying ourselves.
Those floated chipboard floors are terrible, I've come across quite a few that over time have failed badly. I have one that I'm looking at right now! I would suggest you remove that subfloor get to the base and start again.
Sounds like your floor has too much movement & isn't suitable for these large heavy tiles. I seen this problem a few times when there layed on wooden floors there just to big & heavy even when they're laid on a crack mat ! Maybe smaller tiled would be more suitable or even better a Luxury vinyl tiles like Amtico or Karndean !