Accumulating evidence shows improved syntactic processing after exposure to a rhythmically regular compared to an irregular musical prime, environmental noise, or silence. One potentially shared system between musical rhythm and language processing may be responsible for the construction of hierarchical sequences. Following findings of a shorter-lived rhythmic priming effect in Jabberwocky and more precise neural tracking of linguistic constituents in natural language than in Jabberwocky, the present study hypothesized that (a) hierarchical structure building constitutes a key shared mechanism between rhythm and language processing and (b) semantic information may also play a role in structure building. In three experiments, French-speaking typical adults listened to 32-s rhythmic primes before completing six-sentence blocks of grammaticality judgment on natural language and jabberwocky materials in lab and online. Results showed a heavily reduced priming effect present only in the first sentence after a prime in Experiment 1 (natural language, online) and no priming in effects in Experiments 2 (jabberwocky, online) and 3 (natural language, in lab). Replicating previous results, overall grammaticality judgment d' correlated with performance in a rhythm discrimination task. In two out of three experiments, grammaticality judgment performance correlated with rhythm discrimination. These correlations support the hypothesis of a domain-general cognitive network responsible for hierarchical structure building in rhythm and language processing, but do not rule out alternative accounts. However, the priming data showcase that the rhythmic priming effect is reduced when typical speakers process sentences containing linguistic information available at all levels compared to atypical populations processing natural language or typical adults processing syntactic structures in the absence of lexical semantics, and do not suggest a key role of lexico-semantic information in rhythmic priming. Furthermore, relationships between the rhythmic priming effect, rhythm discrimination, and spontaneous speech synchronization suggest that sensitivity to rhythmic priming may be influenced by several factors.