  Casimir forces can appear between intrusions placed in different media driven
by several fluctuation mechanisms, either in equilibrium or out of it. Herein,
we develop a general formalism to obtain such forces from the dynamical
equations of the fluctuating medium, the statistical properties of the driving
noise, and the boundary conditions of the intrusions (which simulate the
interaction between the intrusions and the medium). As a result, an explicit
formula for the Casimir force over the intrusions is derived. This formalism
contains the thermal Casimir effect as a particular limit and generalizes the
study of the Casimir effect to such systems through their dynamical equations,
with no appeal to their Hamiltonian, if any exists. In particular, we study the
Casimir force between two infinite parallel plates with Dirichlet or Neumann
boundary conditions, immersed in several media with finite correlation lengths
(reaction--diffusion system, liquid crystals, and two coupled fields with
non-Hermitian evolution equations). The driving Gaussian noises have vanishing
or finite spatial or temporal correlation lengths; in the first case,
equilibrium is reobtained and finite correlations produce nonequilibrium
dynamics. The results obtained show that, generally, nonequilibrium dynamics
leads to Casimir forces, whereas Casimir forces are obtained in equilibrium
dynamics if the stress tensor is anisotropic.
