This paper investigates how the topical flow of dyadic conversations emerges over time and how differences in interlocutors' personality traits contribute to this topical flow. Leveraging text embeddings, we map the trajectories of $N = 1655$ conversations between strangers into a high-dimensional space. Using nonlinear projections and clustering, we then identify when each interlocutor enters and exits various topics. Differences in conversational flow are quantified via $\textit{topic entropy}$, a summary measure of the "spread" of topics covered during a conversation, and $\textit{linguistic alignment}$, a time-varying measure of the cosine similarity between interlocutors' embeddings. Our findings suggest that interlocutors with a larger difference in the personality dimension of openness influence each other to spend more time discussing a wider range of topics and that interlocutors with a larger difference in extraversion experience a larger decrease in linguistic alignment throughout their conversation. We also examine how participants' affect (emotion) changes from before to after a conversation, finding that a larger difference in extraversion predicts a larger difference in affect change and that a greater topic entropy predicts a larger affect increase. This work demonstrates how communication research can be advanced through the use of high-dimensional NLP methods and identifies personality difference as an important driver of social influence.