[This is joint work with Conceição Nogueira and M. Lurdes Teixeira.] The semidirect product is a fundamental operation in the theory of pseudovarieties of semigroups. In turn, the pseudovarieties of the form V*D, where D is the pseudovariety of all finite semigroups whose idempotents are right zeros, are among the most studied semidirect products. The concept of tameness of a pseudovariety was introduced by Almeida and Steinberg as a tool for proving decidability of the membership problem for semidirect products of pseudovarieties. The fundamental property for tameness is reducibility.  This property was originally formulated in terms of graph equation systems and latter extended to any system of equations. This talk is concerned with the reducibility property for pseudovarieties of the form V*D. Our main results are transfer type results, which state that if a pseudovariety V is reducible with respect to a certain kind of system of equations, then so is the pseudovariety V*D.