Graham, if I have understood your diagram properly I believe:
1) Some liquid will flow out of the top container into the bottom container due to the effects of gravity (we call it liquid head)
2) As this happens the vapour in the lower container will be squeezed and it’s pressure will increase. Some of the vapour will condense, and the heat of condensation will warm the remaining gas, so it is now gas at a higher pressure.
3) In parallel the opposite will happen to the vapour in the top container – it will expand, cool and and up at a lower pressure.
4) Soon the pressure difference between the two vapour pockets will be equal and opposite to the gravity effect and the liquid will stop flowing.
5) In the longer term the flow of liquid will continue very slowly from the top container to the bottom one, as the differences in temperature tend even out and the pressures in both vapour spaces try to return to their original values. Depending on the size of the original vapour spaces and the properties of the liquid, the original vapour space could disappear entirely.
Please do not try this at home with any sort of pressurised flammable liquids.