People often use a rougher strop for pastes as well or the rougher side of a smooth strop.

I just do as was described above. I do a few laps on linen to make sure the edge is dry and clean before going to the fine leather strop.

After I use a pasted strop I clean the edge with my fingers or by stropping on my leg for a second and then go to the linen strop (to make sure all of the paste is off) before going to the leather.