Strange. I have not had your issues on our '06 XK. The two front windows have the auto up feature which is nice. However, the baggage that comes with auto up is that it has to have an obstruction sensor. If the window senses an obstruction, then it stops going up and goes down. This is a safety requirement geared toward kids and their hands, or worse, their heads getting caught by the moving window. I expect that it just senses motor voltage required for closure and if it goes above the threshold, then it assumes that there is an obstruction. This is similar to a garage door opener, or closer if you prefer. It may just be binding rather than an obstruction.
If I were you, then I would take the door apart and lubricate the window regulator with some white lithium grease. I would check the weather stripping to make sure that it is not obstructing the glass path. Worst case, you may need a new window regulator. Those are easy to replace and are generally cheap. I have already replaced two regulators on the WJ and all four on our sedan.
It could be a switch, but my money is on the regulator, the weather stripping, or the regulator motor just going bad.
'06 XK Limited, 5.7L, QD II, Black, 136K
'02 WJ Overland, 4.7L HO, QD, Inferno Red, 108K