I recall visiting Egypt a decade ago, and seeing many dovecotes(?), beehive-shaped structures a dozen feet tall, studded with bird-sized openings, sitting on the (flat) roof of many houses in the countryside. I assume this is fairly traitional in that area of the world.
Unlike something like the Ring-necked Pheasant, North America already has a ton of native pigeons and doves which are superficially pretty similar to their Old World cousins. We do have introduced species, but most came via pet escapees.
The most enterprising is the Eurasian Collared Dove that from a group of ~50 escapees from the side effect of an aviary burglary in the Bahamas in 1974 have now spread right across pretty much all of North America. They arrived local to me in the early 2000’s and while not at all ubiquitous, they are now established here. Interestingly I can anecdotally back up an explosive increase when they first arrived, followed by a decline to a smaller and presumably more stable population.
ETA: Common Wood Pigeon vs native Band-tailed Pigeon
This is exactly what I have experienced with the euro dove. It has roughly coincided with a dramatic increase in coopers hawks in my area. The coopers hawk population seems to have gone down some in the last 5 years or so. and so has the euro dove population. This is strictly based on my personal experience.