<p>In this article, we considered a class of composition operators on Lebesgue spaces with variable exponents over metric measure spaces. Taking advantage of the compatibility between the metric-measurable structure and the regularity properties of the variable exponent, we provided necessary and sufficient conditions for this class of operators to be bounded and compact, respectively. In addition, we showed the usefulness of the variable change to study weak compactness properties in the framework of non-standard spaces.</p>