  We initiate and develop a framework to handle the specialization morphism as
a filtered morphism for the perverse, and for the perverse Leray filtration, on
the cohomology with constructible coefficients of varieties and morphisms
parameterized by a curve. As an application, we use this framework to carry out
a detailed study of filtered specialization for the Hitchin morphisms
associated with the compactification of Dolbeault moduli spaces in [de-2018].
