When you change the mud weight its done by adding weighting...

When you change the mud weight its done by adding weighting agent (increase MW) or diluting the mud (decrease MW) in the surface pit system. The new mud is then circulated down the hole and back to surface so the entire system is now at the new mud weight.

The new higher weight mud now fills the entire wellbore and exerts hydrostatic pressure against the entire wellbore wall and bottom of the hole.

If you're drilling with 9.5ppg mud and the well is nicely balanced and you then hit a zone where formation pressure has increased to close to 9.5ppg EMW (equivalent mud weight)...the mud weight will now be closer to formation pressure and the well will be closer to being underbalanced. There will be signs such as increasing background gas and trip and connection gas peaks that a warning signs of the latter.

You must then increase mud weight or the well will eventually kick on you. Increasing the mud weight restores the balance over the lower overpressured zone but the normally pressured zone above the pressure ramp up depth will now be overbalanced and invasion (and diff sticking) will occur. Unfortunately its a situation where youre between a rock and a hard place...well safety takes precedence over reducing of invasion and wellbore stability.