The kid's finally getting over her first cold, which has produced several sleep-deprived nights for all concerned. With a stuffed up and/or runny nose, she had a really hard time getting to sleep. Several times, she even seemed to panic a bit (understandably) at the prospect of not breathing. She'd always calm down and fall back asleep on our shoulders, but it wouldn't last.
Finally, at 4:30 one morning, we tried putting her in her car seat, so she could breathe a little easier, and it worked like a charm.
No one recommends putting a kid to sleep all night in one [update: see comments. And of course, we let the kid sleep for 10 hrs on the flight to Japan.], but for a couple of hours comfortable nap, it's apparently fine. In fact, she's asleep in it right now.