The investigation of $C^*$-algebras and Hilbert $C^*$-modules with respect to the classical, the weak and the uniform weak Banach--Saks properties is completed giving a full picture, in particular in the non-unital cases. This way some open questions by M. Kusuda and C.-H. Chu are answered. Criteria and structural characterizations are given. In particular, the weak and the uniform weak Banach--Saks property turn out to be invariant under strong Morita equivalence for non-unital $C^*$-algebras.