Given graded C*-algebras A and B, we define the notion of an admissible pair (&phis;, D ) for A and B. Associated to an admissible pair (&phis;, D) is an equivalence class of asymptotic morphisms from A to B. Under certain hypotheses, we develop a composition formula for the composition of asymptotic morphisms arising from admissible pairs. Let Φ = (&phis;, D) be an admissible pair for (A, B) and Ψ = (ψ, E) be an admissible pair for (B, C). We formulate conditions under which (ψ o &phis;, ψ(D) + E) is an admissible pair for (A, C) and the composition [[Ψ]]_a o [[Φ]]_a is equivalent to [[ψ o &phis;, ψ( D) + E]]_a. We then discuss the perturbation theory of regular operators on Hilbert modules. In particular, we prove an analogue of the Kato-Rellich theorem for regular, self-adjoint operators on a Hilbert module. This provides a framework to perform the arithmetic with regular operators needed in the composition formula. We conclude with an application of the composition formula to the stability of K-homology classes of elliptic operators on closed, compact Riemannian manifolds under zeroth order perturbations.