Start with the string
Fresh strings stretch. Poor winding can slip. A damaged string can misbehave all by itself. Rule out the cheap and obvious thing first.
Listen to the nut
Pinging while tuning can indicate binding. If pitch changes after a bend and then snaps back later, friction deserves attention.
Check the bridge
On tremolo-equipped guitars, the bridge needs to return consistently. A floating system also responds to tuning and string-gauge changes as a whole.
Check hardware
Loose tuner hardware, saddle screws and other mechanical movement can create instability or rattles that feel like setup problems.
Separate tuning from intonation
Tuning instability means pitch moves over time or after playing. Intonation means fretted pitch disagrees with the open-string scale. They are related to pitch but are not the same problem.
Record the baseline, change one thing, retune, and test again.