We use the latest cosmic microwave background (CMB) observations to provide updated constraints on the dark matter lifetime as well as on p-wave suppressed annihilation cross sections in the 1 MeV to 1 TeV mass range. In contrast to scenarios with an s-wave dominated annihilation cross section, which mainly affect the CMB close to the last scattering surface, signatures associated with these scenarios essentially appear at low redshifts (z less than or similar to 50) when structure began to form, and thus manifest at lower multipoles in the CMB power spectrum. We use data from Planck, WMAP9, SPT and ACT, as well as Lyman-alpha measurements of the matter temperature at z similar to 4 to set a 95% confidence level lower bound on the dark matter lifetime of similar to 4 x 10(25) s for m(chi) = 100 MeV. This bound becomes lower by an order of magnitude at m(chi) = 1 TeV due to inefficient energy deposition into the inter-galactic medium. We also show that structure formation can enhance the effect of p-wave suppressed annihilation cross sections by many orders of magnitude with respect to the background cosmological rate, although even with this enhancement, CMB constraints are not yet strong enough to reach the thermal relic value of the cross section.