Is it happening on all of them?
I could be wrong, but I think the fit under that big screw is something like a square peg & square hole, meaning tightening won't do anything to help... unless you check the screw and it's currently loose. If it's already snug, don't tighten it further!
Traditionally, players of wooden-peg instruments like violins/violas/cellos will tune upward to reach the target pitch, and never downward. The reason is, when tightening things, one can know that all the parts are snug due to increasing tension, while something might be slightly "caught" when you're tuning downward, later slipping and making the string go flat.
Also, a known culprit of tuning issues is the nut. If you've changed strings and your new strings are even a different brand at the same gauges as the last set (and for a tight-fitting nut, even the same brand strings!), you could end up having the nut slots grab at your strings, making tuning a sporadic hassle. Might be worth investigating further before taking tools to the tuning machinery, IMHO.