In the path integral formulation of quantum mechanics, the amplitudes of all possible paths from starting point A to end point B are added up.
For a free particle, how can all these paths be physically possible? If the particle is to conserve momentum, it must travel in a straight line. It is not interacting with anything else, so the path cannot be a curve in flat spacetime. Why does the integral include physically impossible paths?
First of all, in the path integral we integrate over all off-shell virtual paths. The virtual paths came from how we derived the path integral from the operator formalism by inserting infinitely many completeness relations, see any decent textbook on the matter.
In the semiclassical limit $\hbar\to 0 $, the path integral is dominated by contributions from on-shell classical paths, i.e. solutions to Euler-Lagrange (EL) equations, cf. e.g. this Phys.SE post.
In QM with an external potential, the paths don't necessarily conserve momentum, due to external forces. However, in the free case [which OP seems to be asking about] the classical [but not necessarily the virtual] paths conserve momentum.
By the way, in QFT in the Fourier momentum representation of spacetime [which OP doesn't seem to be asking about] the momentum conservation of off-shell virtual paths is typically a consequence translation invariance of spacetime.
