Every time I drive with the button on on my auto transmission, I get this weird thing where the car kinda "lurches" when coasting to a stop. For example, if I'm coming up to a red light and I let go of the gas to slow down without pressing the brakes, it feels like the car does a rough downshift once...kind of like stepping hard on the brakes for a second and then letting go. This doesn't happen with the button off, when it's off it coasts smoothly to a stop.

 

My 07 XK does the same thing, but in regular drive. I don't have a sport button per say, just a sport selection on the drive selector.

 

I have noticed this behavior with the sport mode engaged as well. I attribute it to the fact that the sport mode also seems to instruct the transmission to downshift a bit sooner. This means the car is downshifting at slightly higher road speeds than normal, which feels a little rough like a manual transmission shift.

 

It's not normal, that's for sure. Buy or borrow a code reader to see if there are any codes present. I wouldn't just ignore it. The sport mode uses different pressures and shift points but everything mechanical is the same as regular mode. Be proactive.