Sequential, multiple assignment, randomized trials (SMART) are a design which allows rigorous comparison of sequences of treatment decision rules tailored to the individual patient, i.e., dynamic treatment regime (DTR). The standard approach to analyzing a SMART is intention-to-treat (ITT) which may lead to substantially biased estimates of DTR outcomes in the presence of partial compliance. For example, partial compliance is a problem in substance use disorder trials such as the ENGAGE SMART. Principal stratification is a powerful tool which stratifies patients according to compliance profiles. Current methods are limited to the single-stage setting. We fill the current methodological gap by developing a rigorous principal stratification framework that leverages a flexible Bayesian non-parametric model for the compliance distribution and a parametric marginal structural model for estimating the mean DTR outcomes in compliance classes. We extend current methods to the multi-stage, SMART setting. We demonstrate the validity of our method through extensive simulation studies and illustrate its application on the ENGAGE SMART.