Well climb and descent wouldn't matter but altitude certainly would.
Hypoxia begins to set in for us at 15,000 feet. I can tell you that I've had to use oxygen on my 18,000 foot skydives. And we weren't even above 15,000 all that long. There have been times when we've overshot the drop zone, and had to circle the airfield for 15 minutes AT 15,000 feet. I haven't so much been hypoxic as I have noticed that by the time we got back around, I was breathing harder... So, another 15 minutes or so, and I might have been.
Their air sacs are more efficient than our lungs are, but I still wouldn't take a bird to or above 15,000 ever, and 12,000 and below is probably preferable. In fact, I'd stay at 10,000 if I could.
As I recall, pressurization systems are set at 8,000 feet, are they not?! So, that is an altitude that we know is absolutely always safe. (Unless there are thunderstorms between point A, and point B. And if I try to go over them I kill my bird.) But I'd absolutely cut it off at 12,000 if I had a bird with me.
The guy who skydives with the hawk routinely takes that bird to 12,000... and they freefall together.
So, 12,000 appears to be okay. But I am just giving you my best guess here. I've never done it myself, nor do I know anyone who has.
(But if I had a plane, it sounds like something I'd do.)
[ame="https://www.youtube.com/watch?v=-vfLDIvBGXE"]Willing To Fly - Peregrine Falcon - YouTube[/ame]