I made an observation a few months back concerning multiple species of neotropical cichlids being in the same tank. In case some new members don't recognize "neotropical," it refers to species native to the tropical areas in the new world--South and Central America, Mexico and southern USA).
None of these cichlids live together in nature. That is a profound point, because it means they never come into conflict with other similarly-territorial minded cichlids. And that alone has a significant impact on their well-being.
Most of them should never be combined with other cichlid species. Exception is some of the dwarf species, in suitably-sized tanks. But as Neale Monks has advised, keeping Firemouth Cichlid(s) in with any of the more robust cichlids is cruel; the Firemouth is a much "gentler" cichlid and deserves better. As just one example, not suggesting this species is mentioned here.
The East African rift lake species is a totally different ballgame, as other members have previous well explained in this thread.