English-German

There is no single German word to cover the many shades of meaning of "abusive", or "abuse" - one of my pet peeves is when news agency reports translated from an English source use "Missbrauch" when it is clear from the context that they should have used "Misshandlung".

So it all depends on the context.
If a person becomes abusive by insulting others, "ausfällig" werden would indeed be a good translation, or you could do a looser translation around "beleidigen".
If it means that the person starts to physically hit others, you could use "gewalttätig werden", "angreifen" etc.

An "abusive relationship" could be translated as "eine von häuslicher Gewalt geprägte Beziehung". When talking about such relationships in general, you will often find the expression "Gewalt in der Beziehung" in German.

An abusive charakter could be a "zu Gewalttätigkeit neigende" person.

Those were exactly the translations I could only come up with. I was wrecking my brain all day, but it seems I was right that there isn't an equivalent. Indeed "Misshandlung" is something very different than "Missbrauch".