the problem with spreading terraforming across the cosmos is the amount of materials you’d have to drag along. this makes it almost prohibitive.
that being said, it’s possible ‘seed pods’ could be sent out, pods that would find a suitable location and start the process.
of course, it’d be very wasteful to spew seed pods across the universe, hoping they might find a suitable planet.
instead, the builders would probably use the gate system to transport the seed pods to planets found to be suitable by the cheaper, more expendable, searcher drones.
Could "seed pods" be transported by comets?