  We present a stochastic lattice theory describing the kinetic behavior of
trapping reactions $A + B \to B$, in which both the $A$ and $B$ particles
perform an independent stochastic motion on a regular hypercubic lattice. Upon
an encounter of an $A$ particle with any of the $B$ particles, $A$ is
annihilated with a finite probability; finite reaction rate is taken into
account by introducing a set of two-state random variables - "gates", imposed
on each $B$ particle, such that an open (closed) gate corresponds to a reactive
(passive) state. We evaluate here a formal expression describing the time
evolution of the $A$ particle survival probability, which generalizes our
previous results. We prove that for quite a general class of random motion of
the species involved in the reaction process, for infinite or finite number of
traps, and for any time $t$, the $A$ particle survival probability is always
larger in case when $A$ stays immobile, than in situations when it moves.
