Language is passed across generations through cultural transmission. Prior experimental work, where participants reproduced sets of non-linguistic sequences in transmission chains, shows that this process gives rise to two characteristic statistical properties of language that enhance its learnability: the statistical coherence of words and the Zipfian distribution of word frequencies. In this study, we extend this work in three ways. First, we replicate and strengthen previous findings using a browser-based experimental procedure with a smaller dataset, demonstrating the robustness of these findings and creating a methodological platform for future research. Second, we show that learners are sensitive to the sequence information that emerges through cultural transmission by showing that reaction times are faster for higher transitional probabilities. These findings suggest that the learning of fine-grained sequence information drives the emergence of statistically coherent units with a Zipfian frequency distribution. Third, we ask whether another cross-linguistic property of language, Zipf's law of Abbreviation, emerges over cultural transmission. We find that the law is present in the sets produced by participants but that it does not evolve over transmission. We discuss how these findings support the proposal that production pressures alone may be sufficient to explain the consistently weak frequency-length correlation observed in natural language.