We discuss the problem on the connectedness of various webs of lattice polytopes by introducing a geometric point of view from the toric Mori theory. To this end, we provide a combinatorial description of toric Sarkisov links in terms of certain sets of lattice points, which we call primitive generating sets. In two dimensions, the description is further translated into the language of lattice polygons. As an application, we prove in two ways (constructive and non-constructive) that reflexive or terminal polygons form a single connected web via inclusion relations even without taking modulo unimodular equivalences.