They would have been imported from regions like Asia Minor/Africa where lions would have been native.

It isn't a myth at all; lions were used in the Colosseum as well as a variety of other exotic beasts like zebras. There were entire systems of mechanisms below the ground in the Colosseum to contain these animals. On the other hand, they wouldn't usually have been 'fought' by gladiators as such, the men being thrown to the lions and bears would have "no defensive armour" (Seneca Letters 7.4) and would have been effectively marked for death. Seneca saw it as glorified murder.