We study the simultaneous filling and embedding problem for
a CR family of compact strongly pseudoconvex CR manifolds of
dimension at least 5. We also derive, as a consequence, the normality of the Stein fibers of the filled-in Stein space under the
constant dimensionality assumption of the first Kohn-Rossi cohomology group of the fiber CR manifolds. Two main ingredients
for our approach are the work of Catlin on the solution of the ∂-equation with mixed boundary conditions and the work of Siu
and Ling on the study of the Grauert direct image theory for a
(1,1)-convex-concave family of complex spaces.