I thought that Lanfear died falling into the finnland doorway. Wasn't there a prohibition to anything related to the shadow? Or, maybe as they fell Moiraine actually killed her.

Either way, because Lanfear died the finns couldn't hold her, and the DO could transmigrate her soul. Moiraine on the other hand is of the light, but she'd come before, using up her "allowed" visits, so the Finns could imprison her -- it was outside the compact.

This doesn't explain how Lanfear became weaker, only why she was transmigrated while Moiraine was imprisoned.

I get the sense BS is being cagey about admitting Lanfear is Cyndane because there's a surprising reveal there, that somehow it's not a direct transmigration (as in Aginor to Dashiva or Belthamel to Halima). Can't wait to find out.