  The transfer matrix in lattice field theory connects the covariant and the
initial data frameworks; in spin foam models, it can be written as a
composition of elementary cellular amplitudes/propagators. We present a
framework for discrete spacetime classical field theory in which solutions to
the field equations over elementary spacetime cells may be amalgamated if they
satisfy simple gluing conditions matching the composition rules of cellular
amplitudes in spin foam models. Furthermore, the formalism is endowed with a
multisymplectic structure responsible for local conservation laws.
  Some models within our framework are effective theories modeling a system at
a given scale. Our framework allows us to study coarse graining and the
continuum limit.
