Now when you say “you tried with bones” what do you mean ? What kind of constraints did you use ? …
If you haven’t tried already try using the stretch to constraint . Just add a bone for each boob and add the stretch to contraint and a target bone for the constraint . Assign all the applicable boob vertices to the bone with the constraint … and I think you will wind up with the results you want … The stretch to constraint maintains the volume of the assigned vertex group so is ideal for organic stuff as well as squash and stretch “toony” applications .
To animate said boobs just key the target bones .
… Gee hope your old enough to play with digital boobs …
If the effect isn’t what you want try doing a search on this forum … I remember there was a post about this not that long ago … And yes the key word you’ll have to use is “boobs” …
You could also try attaching an empty to each breast as a hook (Ew - that sounds painful!), parent the hooks to the nearest bone (Or vertex parent them to three verts around the breasts) and make an animation offset of ¼ - ½ frame or so.