More probably up to an equilibrium. Everything getting to the cold side will freeze and stay there but it must get there first.
My naive model: atmosphere on the warm side goes up, moves with hot humid air toward the cold side, goes down and back to the warm one with cold dry air.
Less naive model: we have three large convection cells here on Earth between the pole and the equator. Maybe they have several cells there too. That could slow down the transfer.
Furthermore dry is never 100% dry and when there is little water left the transfer will slow down. How much water will be left and for how long, I can't say. There could be the equilibrium I'm thinking about or not.
Other variables: volcanoes melt stuff locally, glaciers flow to lower ground, continents move.
The migration of water to the dark side changes the center of mass for the planet, right? Might that be enough, especially if there are any other bodies around to throw in some chaos, to pull the ice back into the warmer region?
Right, but not that's the same thing I'm wondering about. I'm wondering if the change in mass distribution due to ice migration might set things out of balance such that movement is possible. Further, that could still happen with effective tidal lock. Perhaps it's a "punctuated equilibrium" kind of thing where it's stable for a relatively long period until enough ice builds on the dark side that the planet "rolls" and stays in that position until enough ice melts and migrates again.
