Looking in the upstream bugzilla, I found
https://bugs.freedesktop.org/show_bug.cgi?id=12607
which got "resolved" by changing the "in_dgamouse" setting to 0.
Do you have something like this in your games? Then I wonder if all this
could be related to bugs like #444601.
Considering that Bugzilla Bug 12601 is about problems only in quake3/q3-based, it's probably not the same problem (dosbox is surely not quake-based :) ).
And bug#444601 is about xserver-xorg-input-mouse (that works well for me except extra keys, btw).
P.S. Problem in this bug is in some extra mouse movement events. It might be good idea to track them (
e.g. do they occur from evdev input device or not, and after what exactly they start to appear), but I doubt I can do that.