W126 Stumble at idle

[Antony]

Good day, I have a rather odd problem with a 1990 560SEL.
Cold starts, and will run perfectly all the way up to operating temperature without any problems.
Once the engine is warm, if I shut it off, and restart it while warm (say after stopping at a store) I will then have an intermittent stumble at idle. The engine runs fine while driving along at any speed, but at idle, it will falter every 20-30 seconds for the briefest of times. It will nearly die, then just as suddenly go back to smooth idle. Idle for a few seconds, then stumble again.
To reiterate, this condition only appears on a warm engine restart; first thing in the morning, the car will run perfectly for however long it's on, this only happens on a warm restart.
I am at a loss as to what might be causing this condition. I am looking for some troubleshooting steps that I can take to track down this problem.
Any assistance will be much appreciated.
Thank you,
Antony

[pawoSD]

Leaking vac lines, bad idle control valve would be good things to check first. Could also be the air flap/idle closed loop switch, if those malfunction the idle will be unstable.

Sounds like what it is doing is switching from mechanical-run (very low idle) to computer controlled.