  We deal with two following classes of equilibrium stochastic dynamics of
infinite particle systems in continuum: hopping particles (also called Kawasaki
dynamics), i.e., a dynamics where each particle randomly hops over the space,
and birth-and-death process in continuum (or Glauber dynamics), i.e., a
dynamics where there is no motion of particles, but rather particles die, or
are born at random. We prove that a wide class of Glauber dynamics can be
derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the
convergence of respective generators on a set of cylinder functions, in the
$L^2$-norm with respect to the invariant measure of the processes. The latter
measure is supposed to be a Gibbs measure corresponding to a potential of pair
interaction, in the low activity-high temperature regime. Our result
generalizes that of [Finkelshtein D.L. et al., to appear in Random Oper.
Stochastic Equations], which was proved for a special Glauber (Kawasaki,
respectively) dynamics.

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