It's an automatic. This could possibly happen if your facing material on your damper inside your torque convertor wore off but it should still move before the torque convertor goes into lock-up. However if you let it idle or give it very little gas it should not go into lockup and would still move a little bit.
My guess is it's something in the trans, most likely related to your fluid and seals. It's getting cold this time of year and what is happening is that once your fuild warms up it's seeping through seals and loosing pressure inside the transmission. Check your fuild level and possibly change your fuild out, make sure you have the correct amount in. Your torque convertor runs off of automatic transmission fuild so if you don't have enough in there is won't work.