For in that sleep of death what dreams may come.

[William Cloud Hicklin]

I guess they [Dwarves] were reborn as babies, or it seems so logically from the narration in the Appendices: it was not so that a fully grown Durin VII. would all of a sudden appear among people.

Apparently not:

The Dwarves add that at that time Aule gained them also this privilege that distinguished them from Elves and Men: that the spirit of each of the Fathers (such as Durin) should, at the end of the long span of life alotted to Dwarves, fall asleep, but then lie in a tomb of its own body, at rest, and there its weariness and any hurts that had befallen it should be amended. Then after long years he should arise and take up his kingdom again.

--HME XII.383

and

... the reappearance, at long intervals, of the person of one of the Dwarf-fathers, in the lines of their kings - e.g. especially Durin - is not when examined probably one of rebirth, but of the preservation of the body of a former King Durin (say) to which at intervals his spirit would return.

-- ibid.