I'm almost certain that this is Origin by Stephen Baxter.
The planet to which the modern humans were transported was a red moon, which had replaced Earth's original moon.
The slavers were referred to as "zealots" and they appeared to be modern humans, but it is later revealed that they still have tails.
The less advanced hominids were referred to as "hams" (neanderthals), "runners" (homo erectus), "nutcracker men" (non-aggressive hominids resembling orang utans) and "elf men" (?) (aggressive hominids resembling chimpanzees).
The parallel Earth from which the advanced gigantopithecus descended hominids originated had no moon and rotated much faster than our Earth. Individuals of this species were referred to by greatly shortened abbreviations of their personal names, their full names being composed of the names of their ancestors for thousands of generations.