We present two different interacting dark sector models: one in which the dark matter particle is charged under a non-abelian dark gauge group, whose gauge bosons constitute a dark radiation component; and one in which a fraction of the dark matter has efficient number-changing self-interactions that keep it warm. We find that in general the structure formation is slowed down in these models, which addresses a discrepancy in the measurement of the σ8 parameter of large-scale structure. We also perform fits to cosmological data for a generalization of the non-abelian model (in which only a fraction of the dark matter interacts with the dark gauge bosons) and show that it can ease the current experimental tension in the measurement of the Hubble expansion rate H0.