Engaging in fluent conversation is a surprisingly complex task that requires interlocutors to promptly respond to each other in a way that is appropriate to the social context. In this study, we disentangled different dimensions of turn-taking by investigating how the dynamics of child-adult interactions changed according to the activity (task-oriented vs. freer conversation) and the familiarity of the interlocutor (familiar vs. unfamiliar). Twenty-eight autistic children (16 male; M[subscript age] = 10.8 years) and 20 age-matched typically developing children (8 male; M[subscript age] = 9.6 years) participated in seven task-orientated face-to-face conversations with their caregivers (336 total conversations) and seven more telephone conversations alternately with their caregivers (144 total conversations, 60 with the typical development group) and an experimenter (191 total conversations, 112 with the autism group). By modeling inter-turn response latencies in multi-level Bayesian location-scale models, we found that inter-turn response latencies were consistent across repeated measures within social contexts, but exhibited substantial differences across social contexts. Autistic children exhibited more overlaps, produced faster response latencies and shorter pauses than typically developing children--and these group differences were stronger when conversing with the unfamiliar experimenter. Unfamiliarity also made the relation between individual differences and latencies evident: only in conversations with the experimenter were higher sociocognitive skills and lower social awareness associated with faster responses. Information flow and shared tempo were also influenced by familiarity: children adapted their response latencies to the predictability and tempo of their interlocutor's turn, but only when interacting with their caregivers and not the experimenter. These results highlight the need to construe turn-taking as a multicomponential construct that is shaped by individual differences, interpersonal dynamics, and the affordances of the context.