You know what a misnaged is. Someone who is opposed to chassidus. The chassidim did not give them that name, they gave it to themselves.
In those days many of the misnagdim were motivated by a genuine concern that the new “kat” would turn out like the SZ and Frank cults, and abandon Torah & Mitzvos. That is why they did everything they could to try to destroy it. When they or their children saw that this was not happening, that in fact not only was chassidus not causing people to stop keeping mitzvos, it was causing them to become more diligent in their observance, they stopped being misnagdim. They didn’t necessarily become chassidim but they stopped opposing it. That is why R Chaim Brisker said that by his time (more than 100 years ago) there was no longer any such thing as a misnaged leshem shomayim.
But there were always those misnagdim who were not lesheim shomayim, but were simply mecharcherei riv. Those people continued to be misnagdim to this day.