Not surprisingly when I put in the Einstecklauf into my combination gun point of impact shifts to the left about 3 1/2". I suspect this is the different weight affecting the recoil. I shoot left handed as well.

I'm surprised though that it shifts sideways, rather than vertically.

What are other experiences please.

That change is interesting. I assume the change in POI is for the rifle barrel. Just curious what type and caliber einstecklauf and kind of gun. I've had a number of the 22 magnum in both shorter and longer versions and didn't see any change in my drillings.
I never put one in my BBF's or cape guns, so can't share any experience there. I've not put any full length inserts in any of my guns personally, so have no experience wth them, either.
One thing I've learned, every one is different and what happens often causes me to scratch my head and mumble.
The most frustrating one was a drilling that shifted point of impact every once and a while. It would happe, the stay put for a couple months, then out of the blue, it would hit high or low or left or right by as much as 6 inches at 25 yards. Then it would be fine for a while. i later figured out that it had a bad solder joint, which was not a cheap fix. I doubt your change of POI is that kind of thing, probably just one of those quirky, head scratchers.
Good luck,
Bfly

Someone I know that has one in a drilling in 22 Mag, says that the entire gun shoots off center by about 3 inches if the einstucklauf is fired, but the other barrels are empty.

Mine is a full length inser in 7x65r and it seems to group well. Good to hear that with oth BBFs the rifle barrel does shoot to a diffferent point of aim. It is reapeatable so no great hardship to mark the correct zeros on the scope. I also have a driven boar type scope and this will be zeroed for double rifle configuration.