The partition coefficient for a chemical species across a membrane
Partitioning across a membrane is better known as Osmotic pressure. Think of a tube with a "bag" made out of the membrane attached to it. The more concentrated solution goes inside the bag, the less concentrated solution suspends the bag. The liquid from the suspending liquid then diffuses into the bag causes the fluid level to rise against gravity. The height of the rise yields the Osmotic pressure. So it is the solvent that diffuses not the solute. 

If the solute diffuses through the membrane then it is just a diffusion problem such that the rate of diffusion is related to pore size. Bigger pores give faster diffusion. 

To a chemist a "partition coefficient" would also really seem to apply best to a system at equilibrium. If you're considering drugs in the body then an equilibrium probably isn't obtained. The blood sweeps the drug away to react somewhere else in the body. So the drug is continually transported into the blood. 

The other side of this is something like urea in urine and blood. The kidneys actually concentrate urea from the blood plasma into the urine. However this biological process consumes energy. So this sort of process is different than a simple equilibrium.