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But alaunts were a more recent thing. The alani people didn't call their dogs "alaunts", I'd wager they called them whatever the alani word for "dog" was. The spanish and to a lesser extent the british and french called their big game hunting dogs alaunts around 14-1600 AD. Apparently as a homage to some notorious dogs from the alani people, but only incidentally partly descended from them and by then very different in form and function.
By the way I guess most Spanish people didn´t call their dogs Alaunts either. The dogs "we" know call Alaunt Gentil, due to the French guy who classified certain types, they just called Lebrels, Mastiffs used to guard & defend livestock against wild predators they called Mastins, similar to the English word Mastiff and the name Dogo is used for what the French called Dogue and the Germans called Dogge. The word "Alano" probably doesn´t even come from the word Alaunt and isn´t named after the Alani, at least this is what most linguists say. So i guess we have to be careful with the thought that a word, just because it starts with "Al" is named after the Alani tribes.
Yes alaunt wasn't that widespread or popular of a term, the "alaunts" had many other names in many different regions. I'm not that attached to the word at all but it's what you seem to be using at the moment so I'm working with it. My point is simply that the dogs which were famously called alaunts by the odd person (and alaunt gentils and alaunt veantres and etc) weren't alani dogs, but medieval hunting dogs used at the final stage of a hunt.