It's not just over the mountains there can be problems.
Last year I had 800fpm down (and up) 15nm south of Newcastle. The autopilot tripped as it couldn't keep up. The wind was negligible on the ground but showing 270/40 at 8500ft. We were 40nm east of the Lake district which I presume was causing the problem.
Fortunately it only lasted a few minutes. It took full power and a speed loss of 45kts to maintain altitude.
D.O.