It's a '08 X5 3.0si with 13,000 miles. Manufactured in May '08. N52K, a 3.0 liter naturally aspirated engine.
Yes, the transmission seems to shift smoothly and properly most of the time. The vehicle had the transmission replaced under warranty about a year ago because an error message appeared once and it entered into a limp-home mode. A restart cleared everything and it drove normal, but the dealer's diagnostic machine said to replace the transmission.
When I start the car in the morning (cold start), it doesn't do anything out of ordinary. And it drives fine, like there's no problem at all. But when it's all warmed up and I stop and park somewhere for five, ten minutes and up to four hours, and I restart the engine, it runs noticeably rough for about one to two minutes. At around three minute mark, it gets smoother, but sometimes a little roughness can still be felt via uneven power delivery when my foot is on the brakes and crawling forward ever so slightly. It's as if the engine is pulsating. Most of the time, when I'm stationary, after about two minutes, the engine smooths itself out and idles normally. By the way, the roughness seems to get worst after stopping for one to two hours and restarting.
Even right after a warm start, if I keep the engine rpm over 1000, like driving instantly or keeping the throttle pushed, there's no roughness or at least I can't feel it.
If I drive half a mile and stop, most of the time, the roughness disappears. But sometimes there's that last two percent of unsmoothness remaining.
The trick here seems to be to get the engine temperature back up to full operating temperature again ASAP by driving aggressively, then the engine gets smooth quicker.
Before the cylinder head replacement three years ago, I never had any issue with the engine except for the lifter noise when the ambient temperature was cold. This all started right after the cylinder head replacement.
The funny thing is, when I drove about 300 feet out of the dealer lot after the cylinder head replacement, the car suddenly rocked sideways uncontrollably for a few seconds like it was in a tornado. So I went right back to the dealer but no code found. Although it never again exhibited the roughness in the same magnitude, it was the start of this whole thing.
Hope I answered all your questions. If you have any more questions, please let me know and I will do my best to explain them to you. Thank you so much for reading and your help!