M'ikar ha'din there is no issur of eating meat during the 9 days. The minhag is not to, but technically the issur applies only during seudah ha'mafsekes. R' Yisrael Ya'akov Fischer (quoted in the MB printed with his notes) asks how such a minhag could have gotten started, as it poses a problem of ba'al tashchis. What are you supposed to do with all your fleishig leftovers? Throw them out and waste them? He suggests that perhaps since a child is allowed to eat fleishig during the 9 days it removes the problem.
The gemara (Chulin 17) raises the question of whether leftover meat, which was allowed to be eaten in the midbar without shechita, could be eaten once Klal Yisrael entered Eretz Yisrael and shechita was required. The Rosh comments that this is not just a historical question, but has a nafkah minah l'halacha: if a person took a neder not to eat a certain food from day X, can they eat leftovers of that food that remain from beforehand? Since the gemara concludes l'kula, the Rosh paskens l'kula as well.
(Noda b'Yehudah Mh"T Y"D 64 discusses whether a nazir can drink wine that was prepared before he accepted his vow of nezirus, or be mitamei to a meis that died before he accepted his vow of nezirus. Same idea... maybe.)
At first glance the Rosh is difficult. In the desert, killing an animal was matir its meat. The question of the gemara boils down to whether that status-quo of heter remains in place despite the change in circumstance, or whether it falls off. In the Rosh's case, there was never a matir for anything. The question is not whether the status quo of a matir remains, but rather whether the neder can create a new issur on leftovers. Why should it not?
R' Noson Gestetner (in his shiurim to Nedarim daf 3) suggests a solution based on a yesod of the Avnei Nezer (O.C. 483): shechita is not a one time matir, but rather is a peulah na'nimshechet (to use the Rogatchover's jargon). It's like the act of shechita continuously recurs on the animal and continuously recreates the heter.
Based on this, the gemara in Chulin takes on a completely different meaning. An animal was killed in the midbar. That act is a peulah ha'nimshechet, so the heter continuously renews itself. However, once Klal Yisrael enters Eretz Yisrael, meat has a different matir -- it now needs shechita to be eaten. What good is it if the act of killing the animal is nimshechet if that act no longer serves as a matir? The gemara is not sure -- on the other hand, maybe since this meat is leftover meat, the new requirement does not effect it.
This is exactly the Rosh's argument. Just like we see from the gemara that the meat is not effected by the new oser created by the need for shechita, so too, leftover food cannot be effected by the new oser of a neder.