We discuss a general framework of monotone skew-product semiflows under a connected group action. In a prior work, a compact connected group G-action has been considered on a strongly monotone skew-product semiflow. Here we relax the requirement of strong monotonicity of the skew-product semiflows and the compactness of G, and establish a theory concerning symmetry or monotonicity properties of uniformly stable 1-cover minimal sets. We then apply this theory to show rotational symmetry of certain stable entire solutions for a class of non-autonomous reaction-diffusion equations on R^n, as well as monotonicity of stable travelling waves of some nonlinear diffusion equations in time recurrent structures including almost periodicity and almost automorphy.This is a joint work with Professor Mats Gyllenberg (University of Helsinki, Finland) and Professor Yi Wang (University of Science and Technology of China).