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.
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.
Edit: I found this paper about weather on tidally locked planets https://www.pnas.org/doi/10.1073/pnas.1315215111