CAPM and beta for individual stocks
I would say that we don't generally speaking assume that stock betas are constant. For example the paper Explanations for the Instability of Equity Beta: Risk-Free Rate Changes and Leverage Effects, from 1985 (!), cites at least six other earlier papers that the market risk of securities is not stable.
Many papers use a rolling window CAPM estimation approach (a popular way of allowing betas to vary) exactly because CAPM and more generally factor model betas do not seem to be constant. See for example the JFE paper CAPM for estimating the cost of equity capital: Interpreting the empirical evidence.