Our bathroom has 6"x6" ceramic tiles. I think they are mounted on cement board. I need to re-set 3 of them where we stand in front of the sink.

I can't detect any movement of the floor although I'm sure there is some unavoidable flex or they would not be loose.

I need to re-set them flush. My past attempts at similar repairs elsewhere resulted in the repaired tile being higher. The wife was not happy.

I have a oscillating Dewalt tool, numerous chisels, sanders, etc. Exactly what would be the best method or tool for getting the spot cleaned up to reset the 6" x 6" tile flush and not mess up surrounding tiles?

Once done, I'll do the grout.

If the new tiles were higher the difference must be in the mortar, unless they are natural stone. Natural stone tends to have tiles of different thickness. Clean all the mortar out using chisels or those tools that hold a razor blade. Back butter the new tiles and spread the mortar on the floor. Put the new tile in place. If its too high, squish it down more. If you can't squish it down any more, remove some of the mortar. Use a level to make sure the new tile isn't higher than the existing one before it dries.

Remove the tiles and clean off the backs, if you have new tiles to replace them with all the better, at least it eliminates that part of the process.

A cold chisel with hammer to bust up the mortar that is on the floor, get back to the base, you may have to use a hand sander/grinder to clean up.

Then butter the back of tiles and floor and reset the new tile and grout...

Guess my first question would be, why are the tiles moving...

I have reset numerous pool tiles. Use the oscillating saw to saw the grout lines if the tiles are not loose enough to pry up without breaking.

Once they are up, use the saw again to shave the old mortar off the backer board. Clean the old mortar off the tiles, then spread new mortar, and, yes back butter the tile too. Place tiles and tamp down to the level of the adjacent tiles.

A straight edge may be helpful to get level with the old tiles and grout spacers to get the proper space for the grout lines. Wait until the next day to grout and then again the next day to walk on.

Any suggestions for cleaning the backs and not breaking them?

Most of the tiles I reset, I could just break the mortar off the back with my fingers. For the ones that were not that easy, I found soaking the tiles in water for a few hours made it easier to scrape the mortar off, using a wood chisel or similar tool.

If that doesn't work, I would lay the tiles bottom side up on a towel, and then gently use the oscillating saw like a chisel, letting it do the work as opposed to applying a lot of pressure.