I see what you mean, we've got three floors, (not including the cellar!)and a crib or somesuch in your room as well as the cot in the nursery is very handy, but I didn't carry them around if they were asleep (unless in sling).
If the baby is asleep, I leave them where they are and use the monitor.
The main thing is to have somewhere to put them down on each floor, however you do it, as I say, we used the carrycot on the ground floor.
I kept both my babies in their crib in our room til they were about 7 months,then moved them to the cot in the nursery but yes I believe 6 monthhs is the recommended age, if that's practical with your living arrangements.
If you can only get the moses basket in your room, you may find you need the cot earlier,depending on the size of baby and/or rolling over, tht's why I was suggesting having the small cot in your room.