Additionally, the bill of rights was written in an era when there were no laws about immigration to the US at all. The US had completely open borders at that time, anybody could move there if they chose.

It should also be pointed out that, at the time, cross-Atlantic travel was expensive and few people in North America outside of the US had any interest in immigrating.[1] Additionally, American independence ended the transportation of British prisoners to the colonies and severely limited the amount of indentured servant immigration.

[1] During the colonial era, the bigger problem was the risk of white settlers emigrating to a nearby American Indian community. For example, it's likely that the entire colony of Roanoke did this. Even as late as the 19th century, white children captured and raised by the Comanche almost exclusively chose to continue living as Comanche even after being given the chance to return.