Tiny gremlins are at work again, and it looks like they are trapping characters that in Level 10 Embassies. So if you have a Level 10 DOOP, Omicron or Martian Embassy don’t send any essential characters on any Request missions, as you won’t be able to complete the missions and they will stay trapped until TinyCo push a fix.
If your Embassies aren’t at Level 10, you can delay levelling them up by not collecting Doop once you’re close to the Level 10 upgrade requirement of 16455 Doop Credentials collected.

So your options are simple, if you’ve not got a Level 10 Embassy, hold off upgrading or if you want to upgrade or have already either send only non-essential characters on Requests missions, or delay starting any Request missions until TinyCo push a fix. TinyCo are aware of the issue so hopefully a fix will be on its way very soon.
~ Russian Tigger