My 740 with a ZF transmission has a "reaction rod" between the shifter and the transmission in addition to the linkage. It keeps the shifter lined up with the transmission. I don't know if yours is the same as mine, but I was missing a bushing at the transmission end and my shifter moved when driving the car.
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Al Hagenberger '84 245 310K, '87 744 172K

Check the easy stuff first. I would look at the transmission mount, as you said, for the moving shifter. That can be replaced in about 20 minutes. The hard shifts could be caused by a binding kickdown cable. Lube the kickdown cable liberally with a good spray synthetic lube (remove it from its bracket and hold it vertically, spraying a little at a time into the end; work the cable in and out. Repeat until you notice improvement).

Hope this helps.

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Guid ('88 244DL 231k, '92 745 180k)

I suspect, depending on the degree of shifter movement, you may have worn engine mounts. Look between the oil pan and the crossmember....less than 1/4" or touching, replace the mounts. On an older car with who knows how much oil or trans fluid slopping around down there, consider replacing the trans mount too.

Bad news is the shifter mechanism may have to be replaced in toto. I tried to rebuild one and it helped some, but putting in the new assembly ($200 about two years ago) was a revelation. Better than new. You'll need a new shifter button that snaps onto the detent shaft after you've put the new assembly in. Set the shifter linkage correctly too. Read the manual.

Rough 1-2 shift often result of worn check ball in valve body, (see FAQ) as well as too-tight downshift cable.

Good luck.