  We search for integrable boundary conditions and their geometric
interpretation as $D$-branes, in models constructed as generalized
$\lambda$-deformations of products of group- and coset-spaces. Using the
sigma-model approach, we find that all the conformal brane geometries known in
the literature for a product of WZW models solve the corresponding boundary
conditions, thus persisting as integrable branes along the RG flows of our
sigma-models. They consist of the well known $G$-conjugacy classes, twisted
$G$-conjugacy classes by a permutation automorphism (permutation branes) and
generalized permutation branes. Subsequently, we study the properties of the
aforementioned brane geometries, especially of those embedded in the
backgrounds interpolating between the UV and IR fixed points.
