I'm curious why you believe the pigeon to be a Birmingham Roller? I raised Birmingham Rollers for many years, and in fact from looking at the photo, as poor as it is, I would guess it to be a roller myself. Rollers have pretty poor homing instinct compared to a racing pigeon.
Racing pigeons have been selectively bred for returning home as fast as possible in all kinds of weather. It is amazing how selective breeding over thousands of generations has created the ultimate athlete. Racing pigeons have returned home from 600 miles on the day of release. People who race pigeons raise many every year and train them hard releasing them from farther and farther each week, getting them in shape for the races. They feed them the best grains and attend to their health and needs in every way possible. In the end only the champions are valued and bred from to raise the next generation. Birds that don't return are no use to the racers. Sad, but true, only through selection of the very best has this breed been created. All the various breeds of domesticated animals have been created from careful selection for specific desired traits.
Do a web search on the Birmingham Roller and you will be amazed at what acrobatic athletes they are. They pause in flight, and then suddenly spin backwards in a blur like a spinning ball. They spin anywhere from snapping off one quick back flip, to spinning backwards many feet downward, then snap out of the roll and return to the flock. It is an amazing sight to see. They are flown in groups, and when one rolls it stimulates the others to roll, so many times several are spinning down together.
They are not near the strong fast flyers that racing homers are. They tend to fly slow and high circling within sight of their home. Take one and release it ten miles from home, and you may never see it again. Fly them when high winds come up, and you would likely lose them.
As far as how to care for those pigeons, if they were born and raised in captivity, they don't have the skills like their feral cousins when it comes to foraging and living in the wild. The reason they are hanging around is they associate people with food. They need a handout if they are going to make it. That, or they need to join a flock of their feral cousins who will show them the ropes. Still, they likely will not make it long. Probably a hawk will get them because they just don't have the street smarts to avoid them.
