Momentum is always conserved. You just have to draw the boundaries correctly.
A ball falling to earth has a time-independent Lagrangian because you neglect the earth's acceleration. The electron orbiting the hydrogen nucleus moves in a fixed potential for the same reason.
"Draw the boundaries correctly" and you can't do physics.
m = P_S / c² * V_fohs
P_S and V_fohs do not exist in the standard model. The gravitational microcuviture of protons/neutrons cause changes in the P_S (static vacuum pressure) that therefor changes the mass. P_S is around 1.3 * 10^26 N/m².
This is from the Basic Structures of Matter - Supergravitation Unified Theory model, not the standard model.