Variable stars in the VVV globular clusters. II. NGC6441, NGC6569, NGC6626 (M28), NGC6656 (M22), 2MASS-GC02, and Terzan10.

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.