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In this paper we present a point-free theory of Whiteheadean style of space and time. Its algebraic formulation, called dynamic contact algebra (DCA), is a Boolean algebra whose elements symbolized dynamic regions changing in time, with two spatio-temporal mereotopological relations between them: stable and unstable contact. We prove several representation theorems for DCAs, representing them in structures arising from products of contact algebras or from products of topological spaces. We also present a decidable quantifier-free constraint logic for reasoning about stable and unstable mereotopological relations between dynamic regions. We consider the paper as a first step in point-free dynamic mereotopology.