  We introduce a family of generalized d'Alembertian operators in D-dimensional
Minkowski spacetimes which are manifestly Lorentz-invariant, retarded, and
non-local, the extent of the nonlocality being governed by a single parameter
$\rho$. The prototypes of these operators arose in earlier work as averages of
matrix operators meant to describe the propagation of a scalar field in a
causal set. We generalize the original definitions to produce an infinite
family of ''Generalized Causet Box (GCB) operators'' parametrized by certain
coefficients $\{a,b_n\}$, and we derive the conditions on the latter needed for
the usual d'Alembertian to be recovered in the infrared limit. The continuum
average of a GCB operator is an integral operator, and it is these continuum
operators that we mainly study. To that end, we compute their action on plane
waves, or equivalently their Fourier transforms g(p) [p being the
momentum-vector]. For timelike p, g(p) has an imaginary part whose sign depends
on whether p is past or future-directed. For small p, g(p) is necessarily
proportional to p.p, but for large p it becomes constant, raising the
possibility of a genuinely Lorentzian perturbative regulator for quantum field
theory. We also address the question of whether or not the evolution defined by
the GCB operators is stable, finding evidence that the original 4D causal set
d'Alembertian is unstable, while its 2D counterpart is stable.
