If I understand what your asking correctly, the "feeling" of the water cooling is probably due to fatigue from your temperature sensors in your skin akin to becoming accustomed to a bad smell. The movement would then give them enough of a rest to enable correct feeling again.

The main effect occurs when the bath water (and of course you) are still enough and the convection currents are weak enough for water surrounding your skin to come to about the same temperature as your skin. It need not be hot water; cool water does the same thing in reverse. The still water heats (or cools) your skin, and in doing so, it cools down (or heats up) to about the new temperature of your skin. Meanwhile, your sensory nerve endings in your skin settle down to accept the new temperature as normal, more or less.

Then, sooner or later, you move, setting the water in motion. The swirls remove the cooler (or warmer) layer of water around your skin, that had been shielding you from the hotter (or colder) water further away. That hotter (or colder) water now reaches your skin, making you well aware that the frying pan (or fridge) is not as hot (or cold) as the fire (or freezer).

In fact, if you move very cautiously and slowly, you can actually feel the swirls of hotter (or colder) water moving over your skin. Personally I find the effect most marked (and least harmful) when I have been sitting and thinking too long and too still in a warm bath that has cooled down to below my comfort levels. The little swirling vortices are so obvious that it almost feels as though one should be able to see them.

Happy soaking (especially in the weather you probably have been having)

Jon