The author considers a family of immersed hypersurfaces in Riemannian manifolds which satisfies the evolution equation \[ \frac{\partial F}{\partial t}(p,t)=W(p,t)N(p,t), \] where \(F:M^{n}\rightarrow\bar{M}^{n+1}\)is the immersion, \(N\) is the unit normal and \(W\) is a suitable function of the principal curvatures of the hypersurface. Some existence results are proved when the initial hypersurface is isoparametric and the ambient space is a space form or a rank-one symmetric space of noncompact type. When \(W\) is an odd function, the author proves the avoidance principle and that embedding is preserved, using Hamilton’s trick.