We consider the notion of input-to-state multistability, which generalizes input-to-state stability to nonlinear systems evolving on Riemannian manifolds and possessing a finite number of compact, globally attractive, invariant sets, and in addition satisfies a specific condition of acyclicity. We prove that a parameterized family of dynamical systems whose solutions converge to those of a limiting system inherits such input-to-state multistability property from the limiting system in a semiglobal practical fashion. A similar result is also established for singular perturbation models whose boundary-layer subsystem is uniformly asymptotically stable and whose reduced subsystem is input-to-state multistable. Known results in the theory of perturbations, singular perturbations, averaging, and highly oscillatory control systems are here generalized to the multistable setting by replacing the classical asymptotic stability requirement of a single invariant set with attractivity and acyclicity of a decomposable invariant one.