We consider special flows over the rotation by an irrational α under
the roof functions of bounded variation without continuous, singular part
in the Lebesgue decomposition and the sum of jumps ≠0. We show that
all such flows are weakly mixing. Under the additional assumption that
α has bounded partial quotients, we study weak Ratner's property.
We establish this property whenever an additional condition (stable under
sufficiently small perturbations) on the set of jumps is satisfied. While
it is classical that the flows under consideration are not mixing, one
more condition on the set of jumps turns out to be sufficient to obtain
the absence of partial rigidity, hence mild mixing of such flows.