This paper introduces a new formalism to describe the acceleration of particles in highly conducting astrophysical plasmas. This method tracks the evolution of particle momenta in a sequence of locally comoving reference frames, where the electric field exactly vanishes. It provides an efficient characterization of acceleration in sub- and ultra-relativistic settings, in Cartesian or non-Cartesian geometries, flat or non-flat spacetime. This paper provides three concrete, original applications: acceleration in a magnetized expanding wind from a compact object, acceleration in the vicinity of a black hole, and acceleration in a relativistic magnetized turbulence.