Well, I watched the struggles for quite some time (sneaking peeks at the cam while working). Not sure if the feather got dislodged or was swallowed. He/she was trying to do both but I missed the final outcome. Of course, this was the baby with it's back to the camera so said stuck feather could still be in place! How traumatizing for the poor thing... or maybe more so for the viewers? Sure seemed like it was crying out to its siblings "Help me get this thing out!" 
Feathers and legs and chicks are like little kids putting peas up their nose, an annual event. The falcons don't swallow their food all the way to their stomach, it does make a pitstop in the crop which is (simplified version here) about half-way down their throat so the feather/leg can't go very far. Also, these birds have an amazing regurgitation reflex, remember they hack up pellets (such an nice innocuous name for the peregrine version of a hairball) all the time when they begin eating feathers etc on a regular basis. At the moment their parents of providing them with just the meat but soon they will be trying to eat everything/anything.
So while it looks like they are in great distress, it is a common occurrence and mostly they work it out. Has it ever happened that we remove accessories from their mouths, yup, banding in Brandon a few years ago, one chick was yelling at me (and the Brandon Sun photographer) with a foot sticking out of her mouth - hard to take her seriously with this Wiley Coyote thing going on. We carefully removed it so she could play big bad birdie again.
At the 1430h feeding, all the chicks were eating and there was no sign of the obstinate feather that I could see ... so looks like the little darling managed to work out a solution to its problem on its own ...
