  The Galactic globular clusters (GGCs) located in the inner regions of the
Milky Way suffer from high extinction that makes their observation challenging.
The VVV survey provides a way to explore these GGCs in the near-infrared where
extinction effects are highly diminished. We conduct a search for variable
stars in several inner GGCs, taking advantage of the unique multi-epoch,
wide-field, near-infrared photometry provided by the VVV survey. We are
especially interested in detecting classical pulsators that will help us
constrain the physical parameters of these GGCs. In this paper, the second of a
series, we focus on NGC6656 (M22), NGC6626 (M28), NGC6569, and NGC6441; these
four massive GGCs have known variable sources, but quite different
metallicities. We also revisit 2MASS-GC02 and Terzan10, the two GGCs studied in
the first paper of this series. We present an improved method and a new
parameter that efficiently identify variable candidates in the GGCs. We also
use the proper motions of those detected variable candidates and their
positions in the sky and in the color-magnitude diagrams to assign membership
to the GGCs. We identify and parametrize in the near-infrared numerous variable
sources in the studied GGCs, cataloging tens of previously undetected variable
stars. We recover many known classical pulsators in these clusters, including
the vast majority of their fundamental mode RR Lyrae. We use these pulsators to
obtain distances and extinctions toward these objects. Recalibrated
period-luminosity-metallicity relations for the RR Lyrae bring the distances to
these GGCs to a closer agreement with those reported by Gaia, except for
NGC6441. Recovered proper motions for these GGCs also agree with those reported
by Gaia, except for 2MASS-GC02, the most reddened GGC in our sample, where the
VVV near-infrared measurements provide a more accurate determination of its
proper motions.
