  Interweaving relations are introduced and studied here in a general Markovian
setting as a strengthening of usual intertwining relations between semigroups,
obtained by adding a randomized delay feature. They provide a new
classification scheme of the set of Markovian semigroups which enables to
transfer from a reference semigroup and up to an independent warm-up time, some
ergodic, analytical and mixing properties including the $\varphi$-entropy
convergence to equilibrium, the hyperboundedness and when the warm-up time is
deterministic the cut-off phenomena. We also present several useful
transformations that preserve interweaving relations. We provide a variety of
examples of interweaving relations ranging from classical, discrete, and
non-local Laguerre and Jacobi semigroups to degenerate hypoelliptic
Ornstein-Uhlenbeck semigroups and some non-colliding particle systems
