We study multi-parameter deformations of isolated singularity function-germs on either a subanalytic set or a complex analytic space. We prove that if such a deformation has no coalescing of singular points, then it has weak constant topological type. This extends some classical results due to Lê and Ramanujam (Am J Math 98:67–78, 1976) and Parusiński (Bull Lond Math Soc 31(6):686–692, 1999), as well as a recent result due to Jesus-Almeida and the first author (Int Math Res Notices 2023(6):4869–4886, 2023). It also provides a sufficient condition for a one-parameter family of complex isolated singularity surfaces in C3 to have weak constant topological type. On the other hand, for complex isolated singularity families defined on an isolated determinantal singularity, we prove that μ -constancy implies weak constant topological type.