This article is the third part of a scientific project under the general title "Geometrized vacuum physics based on the Algebra of Signatures". In the first two papers [1,2], the ideal (i.e., non-curved and immobile) local region of vacuum was studied and the foundations of the Algebra of Signatures were laid. This article considers the possibilities of describing the curved and moving state of the same vacuum region on the basis of the mathematical apparatus of the Algebra of Signatures. The reasons for the multilateral consideration of vacuum and twisting of intra-vacuum processes into spiral bundles are disclosed. The 4-tensor is introduced for two-sided and 16-sided consideration of the curvature of the local vacuum region. On the basis of kinematic models, the following assumptions were made: about the inert properties of vacuum layers; about the possibility of displacement of vacuum layers relative to each other at a speed significantly exceeding the speed of light; about the possibility of "rupture" of the local region of vacuum. The proposed kinematic models of the movement of vacuum layers can be a theoretical basis for the development of "zero" (i.e., vacuum) technologies.