In this work, we address charge transfer within complex arrangements of nucleobases from a theoretical and numerical point of view. We study dendrimers constructed from T-shaped double-stranded DNA three-way junctions (3WJs). The electronic structure of these junctions is computed on an atomistic level using a chemically specific Su-Schrieffer-Heeger Hamiltonian, which has been extended by a nonretarded reaction field to take solvent polarization effects into account. Hopping rates through isolated 3WJs are derived by analyzing the emerging potential energy surfaces using Marcus' theory of charge transfer. We find highly anisotropic transport; the corresponding rates are used to compute the global trapping kinetics on DNA dendrimers with a central absorbing core. As a potential application, we discuss a DNA-based drug delivery system.