In recent years, increasing attention has been placed on the development of phylogeny-based statistical methodologies for uncovering site-specific changes in amino acid fitness profiles over time. The few available random-effects approaches, modelling across-site variation in amino acid profiles as random variables drawn from a statistical law, either lack a mechanistic codon-level formulation, or pose significant computational challenges. Here, we explore a simple and fast method based on a predefined finite mixture of amino acid profiles within a codon-level mutation-selection substitution model. Our study detects shifts of amino acid profiles over a known sub-clade of a tree, using simulations with and without shifts over the sub-clade to study the properties of the method. We apply the approach to a real data set, previously studied with other methods: lists of sites identified as having undergone a change in amino acid profile have obvious overlap between methods, while also showing notable differences.