The current main popular implementation of Random Forests (RF) (i.e. the randomForest package) is available only for univariate (continuous or discrete) responses. On the other hand, mixed models are inherently multivariate models, that is models that deal with vector-valued responses. Fortunately, extensions of RF for multivariate responses, in particular for handling longitudinal data, do exist.
LongitudiRF is one of the R packages that implement Random Forests for longitudinal data of which I am aware. A lot more information can be found at this recent review paper on longitudinal data with Random Forests.
Related posts:
How can I include random effects (or repeated measures) into a randomForest
How to deal with hierarchical / nested data in machine learning
Random forest for binary panel data