One way to think about it is a vector of pressure on the aerofoil. To complicate things it also depends on the type of flaps, but generally flaps increase wing area so the centre of pressure will move aft accordingly. It`s reaction on pitch depends on its vector against centre of gravity. With large fowler / slated flaps you may have the centre of pressure move forward and aft of the centre of gravity during deployment as centre of pressure moves aft, then as the flaps extend to a greater deflection causing drag the centre of pressure moves forward again.
Large aircraft use trim compensation systems to help with the changes. Slats also act in the same way moving the centre of pressure forward.
Design of light aircraft means changes in trim and control loading are kept low.