The dog becomes Snuffles, and also becomes theirs wholeheartedly. Over time he gets softer around them; less raised hackles and almost no whimpers. Without the signs of fear he's almost silent around the house and Tonks jokes that they should get some stealth and tracking tips off of him.
(It raises something in their mind, that silence despite the sheer size of the creature and his long claws, something from her childhood and the days when her cousin would come over with his friends. But Tonks is not thinking about her childhood these days. Quite pointedly so.)
Eventually, and all too soon for their tastes, Charlie's summons back to Romania arrives and he suggests, quite genuinely, that Snuffles leaves with him. As though a dragon couldn't eat him in one gulp. As though there wouldn't be a dozen new and scary new people for him to be forced to acclimatise to.
Needless to say, Tonks fights him on this. (And wins).