The driver's window of my E39 is moving veeeery slooowly... any thoughts?

Try this before dismantling anything.... I had this on a car of mine that was about 6 years old, sometime later a friend did too. It turned out in both cases to be the window guides aging and "gripping" the window harder. Lubrication solved the problem, I would not recommend an oil based product. I used Armour All "original" for interiors (also great on exteriors)

as it is formulated for use on rubber andplastics. With the window down spary a "moderate" amount into thelength of the window guides and then operate the window up and down afew times to spread the lubricant. Problem solved.Adrian

Thanks Adrian!

But the E39 has some soft material on those guides (like velour fabric) and I don't think it would work...

I think that is the same material as in the other cars where I have used Armour All. The material looks like rubber coated in velor / suede. I have the same material on my E91 3 series. You're basically looking for something to lubricate the guides which is not oil based as this might perish the rubber. I learned the Armour All trick from a Nissan main dealer about 10+ years ago.

I had to replace two power windows on my E39. They're poorly designed.