  In highly conducting astrophysical plasmas, charged particles are generically
accelerated through Fermi-type processes involving repeated interactions with
moving magnetized scattering centers. The present paper proposes a generalized
description of these acceleration processes, by following the momentum of the
particle through a continuous sequence of accelerated frames, defined in such a
way that the electric field vanishes at each point along the particle
trajectory. In each locally inertial frame, the Lorentz force affects the
direction of motion of the particle, but the energy changes solely as a result
of inertial corrections. This unified description of Fermi acceleration applies
equally well in sub- and ultrarelativistic settings, in Cartesian or
non-Cartesian geometries, flat or nonflat space-time. Known results are
recovered in a variety of regimes -- shock, turbulent and shear acceleration --
and new results are derived in lieu of applications, e.g. nonresonant
acceleration in relativistic turbulence, stochastic unipolar inductive
acceleration and centrifugo-shear acceleration close to the horizon of a black
hole.
