The Lorentz force and Maxwell's Equations gives answers to many physics problems, and the answers given by both methods are consistent.
For example, consider the problem of a conducting rod of length $\ell$ sliding at speed $v$ on two rails, with a $\mathbf{B}$ field normal to the plane of the rod/rails. Lenz's law, which is derived from Maxwell's equations can be used to find $V=-B\frac{dA}{dt}=-B\ell{v}$. On the other hand, consider a charge carrier $q$ in the conducting rod. $q(E+vB)=0$, so $E=-vB$ and integrating over the length of the rod gives $V=-B\ell{v}$. Both Lorentz force and Maxwell's Equations have given the same answer.
However, it appears that Maxwell's equations and Lorentz force appear self contained; Maxwell's equations is not concerned with particles at all. Can one be derived from the other, or is there an overarching structure from which Maxwell's Equations and Lorentz force are corollaries?