So again, in the book I'm writing, and yes, I do understand it is not plausible for humans to fly, I want one of my characters to be hit while flying and continue to fly with broken ribs. Now along the bounds of fantasy (and I still want it to be realistic enough) I'm not sure a bird, though quite different from human anatomy and almost because of that very factor, would even be able to fly with broken wings, pain being irrelevant for the time being. For better context, the wings protrude from the back/shoulder blades as one would picture a modern-day angel. If a bird cannot fly with broken ribs, do you think one of my avian people could? And what kind of damage do you imagine it would cause to continue flying?
Yes... at least in the short term.
In general, major injuries are rarely enough to with absolute certainty stop you from doing what it takes to get to safety. Just in my own personal experience I've known people to jump a fence with a broken back, punch someone with a broken fist, jump with a torn ACL, run on a broken tibia, and walk with a cut Achilles tendon. Injured humans often experience a combination of adrenaline surge and shock which causes us to ignore the pain factor of an injury and continue moving to the best of our physical abilities until we get to somewhere safe enough to tend our wounds.
When you take away pain as a limiting factor, the bodies of most animals are full of redundancy. Every major action is the culmination of many muscles relying on multiple attachment points; so, when one bone is compromised, you generally still have effective adjacent anchor points to compensate. Furthermore, because bones are encased in tendons, ligaments, and muscles that inflame after a break, even when you have no extra bones to work with, the tissues surrounding the bone can in most cases give enough structure to hold the broken bone more or less in place so you can continue to use it for structure (to a degree). This is especially true of your ribs which have very tough and sinewy intercostal muscles connecting each rib to the next. So even when you do break a rib, it is basically splinted to the adjacent ribs by the sinew.
Anatomical Reasons you can fly with broken ribs.
Using shoulder muscles an an analog for how your wing muscle configuration will be, you will see that most of your flight muscles should not attach to the ribs at all. For an angel like birdman to fly, he would likely need some kind of extra layer of muscles going over the arm's muscles and attaching to extra protuberances in the sternum, clavicle, and scapula with only a single extra sheath of Serratus Anterior muscles actually attaching to the ribs. This means that all the major muscles will not directly attach to the ribs at all; so, one or two broken ribs will at most slightly limit the rotational strength of the wings, but not enough to render flying impossible.
So, if you had a major break in the sternum, clavicle, or scapula, then you might lose too much structure in your wing to fly, but the only way you might become anatomically unable to fly from broken ribs would be if you badly broke all of your floating ribs... but even then, limited flight should still be doable with the pectoral attachments in you sternum.
How it could play out.
If your bird man was struck in the chest during a fight for his life (regardless of if he has a human or bird like rib cage), he would likely be able to fly away just fine, but within about 15 seconds to a minute, the adrenaline surge will start to weir off and he'd start to notice a significant weakness in the wing on the injured side and feel like he needs to land. If he's still in danger, he could probably push through this weakness and get in an extra minute or two of flying before the body runs out of adrenalin to push and the weakness becomes overwhelming forcing him to land. Once he lands, his body will quickly become resistant to pushing any further. If he tries to take off again, he'll probably feel too weak to do so. His strength to fly is now gone, but he may not yet realize the severity of his injury because he could still be in shock which could easily last another 30 minutes to an hour... less if he takes the time to try to stop and rest. From the time the shock weirs off to several weeks to months later, your birdman will probably not be able to fly.
It would depend on the severity of the injury, and also on the location.
In particular, flight would be adversely affected if the break is close to the joint where the wing attaches to the body, thus preventing the bird from flapping its wing and generating lift. (Related story about a hawk with an injured coracoid bone.)
But a minor fracture in a less critical location could still allow the bird to fly, albeit slower, lower, and more cautiously to avoid exacerbating the injury.