I would like to know too.

The grout isn't the problem - Your tile was installed improperly over a base with too much flex which is making the stone move and the grout pop.

The true fix is to remove the floor completely and have it installed again properly.

Yup, its not a grout problem. It is what the tiles are sitting on that is the problem. Something is moving underneath them and the weakest point is the grout, so it breaks when it flex's. If you had larger tiles, they'd probably have snapped too. You need to bring up the floor to make repairs. I am not sure whether you can just repair 1 section or the whole floor needs to come up though. May be hard now to match the original grout

The subfloor under the tile needs to be at least 1 1/4" to keep it from flexing. When ever I do a pebble floor I always use an epoxy grout such as Laticrete SpectraLock. If you can get in from a ceiling below you could try and sneak in some more floor joists to take the defection out of the floor. Before I do any tile work thats THE first thing I check. The deflection of the floor. There's a process for doing this as well. It's never just setting tile down...that'd be far too easy : )