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The problem is posed of further extending the axiomatic construction proposedin Part 1 for non-local point transformations mapping in each other differentcurved space times. The new transformations apply to curved space times whenexpressed in arbitrary coordinate systems. It is shown that the solutionpermits to achieve an ideal (Gedanken) experiment realizing a suitable kind ofphase-space transformation on point-particle classical dynamical systems.Applications of the theory are discussed both for diagonal and non-diagonalmetric tensors.

Theory of non-local point transformations - Part 2: General form and Gedanken experiment