So, I will concentrate on Europe. Around year 2000, no flamingo species was yet self-sustaining in EAZA zoos, too few zoos bred them. Greaters were first to reach that status, they seem to be the easiest zoo species and well suited for continental Europe. Caribbeans reached that status within last decade, Chileans are approaching that now I would say despite high mortality of their chicks. Lessers are hard nut to crack, still faaar away but with good steps lately. The two high Andrean species are basicaly doomed.
Some good reagular breeders of Greaters - Ohrada, Dvur Kralove, Olomouc, Zlin, Jihlava, Prague, Bratislava, Vienna, Budapest, Basel, Slimbridge, Bussolengo, GaiaPark last year.
Caribbeans - Ostrava, Dresden, Heidelberg, Slimbridge, Chester, Avifauna...
Lessers - Leipzig, Karlsruhe last year