Children are natural observers and mimics of other's verbal and nonverbal cues. I am wondering if the verbal and nonverbal cues that children internalize has an effect that is not apparent until they reach adolescence and post adolescence.
I am not aware of a research study that answers your specific question, but one place that you might look to get a better idea about children's communicative behavior after divorce is Paul Schrodt and Andrew Ledbetter's work on communication family patterns and mental well-being (http://onlinelibrary.wiley.com/doi/10.1111/j.1468-2958.2007.00302.x/full). Another great article to given you some ideas about children's adjustment after divorce is Joan Kelly's 2000 review piece that collects and analyzes a lot of interesting research (http://www.sciencedirect.com/science/article/pii/S0890856709662948).