The Changelings apparently were persecuted by Solids throughout their history, until they managed to found the Dominion and reverse their fortunes. We could assume the Solids took the Changelings off their birthworld originally, and spread them out as research subjects, curios, or possibly useful domestic animals. Although "Chimera" does establish that Changelings are capable of powered spaceflight, at sublight speeds at least.
The ability to mimic not only people but technology to perfection would probably allow Changeling individuals to rule worlds the way vampires and other clever immortals gain world dominance in mainstream fantasy writing. All it takes is a bit of patience, a bit of sense of drama, and a string of really mysterious deaths...
One would expect there to have been a stage in their history when they weren't a giant planet-sized pool of goo, but rather a series of individual puddles subsisting in "realistic" ecological niches on their birthworld. Then there would follow discovery by sapient Solids, diaspora, and a few bouts of genocide where any large concentrations of goo encountered by Solids would be ruthlessly destroyed. And then the slow ascent to power - and only at the conclusion of that would the Founders be able to gather together again, this time on hideout worlds where they themselves were the only thing around.