Valid surrogate endpoints can be used as a substitute for a true outcome of interest to measure treatment efficacy in a clinical trial. This work is motivated by a trial of a gene therapy where the clinical outcomes are measured repeatedly. We propose a causal inference approach to validate a surrogate by incorporating baseline covariates and longitudinal measurements using a mixed modeling approach, and we define models and quantities for validation that may vary across the study period using principal surrogacy criteria. We consider a surrogate-dependent treatment efficacy curve that allows us to validate the surrogate at different time points, or integrate over multiple time points for an overall measure of surrogate validity. We extend these methods to accommodate a crossover design where all patients eventually receive the treatment. Because not all parameters are statistically identified in the general setting, we rely on informative prior distributions to obtain inference.  We consider the sensitivity of these prior assumptions as well as assumptions of independence among certain counterfactual quantities conditional on pre-treatment covariates to improve identifiability.