Finite-state machine-based scenario-aware dataflow (FSM-SADF) is a dynamic dataflow model of computation that combines streaming data and finite-state control. For the most part, it preserves the determinism of its underlying synchronous dataflow (SDF) concurrency model and only when necessary introduces the non-deterministic variation in terms of scenarios that are represented by SDF graphs. This puts FSM-SADF in a sweet spot in the trade-off space between expressiveness and analyzability. However, FSM-SADF supports no notion of compositionality, which hampers its usability in modeling and consequent analysis of large systems. In this work we propose a compositional semantics for FSM-SADF that overcomes this problem. We base the semantics of the composition on standard composition of processes with rendezvous communication in the style of CCS or CSP at the control level and the parallel, serial and feedback composition of SDF graphs at the dataflow level. We evaluate the approach on a case study from the multimedia domain.