In communication, the behavior you're describing would be called convergence. The key action would involve converging in speech pattern or (more generally) accommodating another's speech pattern.
The concept is a large part of Communication Accommodation Theory, developed in the early 1970s by Howard Giles. It's a framework that predicts factors for people making their speech more similar to a conversational partner (convergence) as well as more different from that partner (divergence). This 2007 entry for CAT in Explaining Communication: Contemporary Theories and Exemplars provides an overview of the theory as a whole.
Within this theory, convergence is a subconscious strategy of adapting to the speech patterns of one's interlocutor. That can be motivated by several factors, including a desire to gain acceptance with the people we're talking to. In contrast, divergence can be a way of maintaining one's ties to a prior identity, like a politician maintaining or even exaggerating the speech patterns of the region they represent when they speak to colleagues with other accents and patterns.
Within this research, converge and accommodate are used as verbs to describe this behavior. For example:
Bourhis, Roth, and MacQueen (1988) found that physicians , nurses,as well as hospital patients considered it more appropriate for health professionals to converge to the patients’ everyday language than to maintain their medical jargon.
This has been observed in a number of settings also where, for example, a travel agent accommodated her pronunciation to the different socioeconomically based language styles of her Welsh clientele (N. Coupland, 1984) and, in Taiwan, where salespersons converged more to customers than vice versa (van den Berg, 1986).
So to take your example sentences, you could say:
Oh, your accent converged with mine.
Come on, your words accommodate mine.