The role played by interfaces in metallic multilayers is not only to change the momenta of incident electrons; their symmetry lowering also results in an enhancement of the effects of spin-orbit coupling, in particular, the flipping of the spins of conduction electrons. This leads to a significant reduction of a spin current through a metallic interface that is quantitatively characterized by a dimensionless parameter δ called the spin memory loss (SML) parameter, the interface counterpart of the spin-flip diffusion length for bulk metals. In this paper, we use first-principles scattering calculations that include temperature-induced lattice and spin disorder to systematically study three parameters that govern spin transport through metallic interfaces of Cu with Pt, Pd, Py (permalloy), and Co: the interface resistance, spin polarization, and the SML. The value of δ for a Cu|Pt interface is found to be comparable to what we recently reported for a Au|Pt interface [Gupta, Phys. Rev. Lett. 124, 087702 (2020)0031-900710.1103/PhysRevLett.124.087702]. For Cu|Py and Cu|Co interfaces, δ decreases monotonically with increasing temperature to become negligibly small at room temperature. The calculated results are in good agreement with currently available experimental values in the literature. Inserting a Cu layer between Pt and the Py or Co layers slightly increases the total spin current dissipation at these compound interfaces.