  We present an analysis of interspecies interactions between Rydberg
$d$-states of rubidium and cesium. We identify the F\"orster resonance channels
offering the strongest interspecies couplings, demonstrating the viability for
performing high-fidelity two- and multi-qubit $C_kZ$ gates up to $k=4$,
including accounting for blockade errors evaluated via numerical
diagonalization of the pair-potentials. Our results show $d$-state orbitals
offer enhanced suppression of intraspecies couplings compared to $s$-states,
making them well suited for use in large-scale neutral atom quantum processors.
