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The Galactic globular clusters (GGCs) located in the inner regions of theMilky Way suffer from high extinction that makes their observation challenging.The VVV survey provides a way to explore these GGCs in the near-infrared whereextinction effects are highly diminished. We conduct a search for variablestars in several inner GGCs, taking advantage of the unique multi-epoch,wide-field, near-infrared photometry provided by the VVV survey. We areespecially interested in detecting classical pulsators that will help usconstrain the physical parameters of these GGCs. In this paper, the second of aseries, we focus on NGC6656 (M22), NGC6626 (M28), NGC6569, and NGC6441; thesefour massive GGCs have known variable sources, but quite differentmetallicities. We also revisit 2MASS-GC02 and Terzan10, the two GGCs studied inthe first paper of this series. We present an improved method and a newparameter that efficiently identify variable candidates in the GGCs. We alsouse the proper motions of those detected variable candidates and theirpositions in the sky and in the color-magnitude diagrams to assign membershipto the GGCs. We identify and parametrize in the near-infrared numerous variablesources in the studied GGCs, cataloging tens of previously undetected variablestars. We recover many known classical pulsators in these clusters, includingthe vast majority of their fundamental mode RR Lyrae. We use these pulsators toobtain distances and extinctions toward these objects. Recalibratedperiod-luminosity-metallicity relations for the RR Lyrae bring the distances tothese GGCs to a closer agreement with those reported by Gaia, except forNGC6441. Recovered proper motions for these GGCs also agree with those reportedby Gaia, except for 2MASS-GC02, the most reddened GGC in our sample, where theVVV near-infrared measurements provide a more accurate determination of itsproper motions.

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