One of my smartest friends noticed that when his support team answered emails quickly, the customer would treat this as an implicit invitation to shift the support thread into a support chat, via email.
They added a 3-hour delay before support sees any email, specifically to prevent threads from becoming chats. Note that phone support is also available; people with time-critical issues are encouraged to call in for immediate help.
The delay has been a huge success because people correctly assign priority to their concerns by selecting the medium. The back-and-forth is more focused and does not get off-track.
An unexpected bonus is that before the delay was introduced, people would often remember how one particular support rep helped them in the past and would hit reply on an old thread to pose a new question, unrelated to the original request. This was confusing (support people leave) and would mess up their issue tracking and happiness metrics.
After the delay, this behaviour went away almost completely and they didn't experience a statistically significant drop in incident satisfaction.
In conclusion, use email for email and use chat for chat. Email starts to feel like chat if you reply too quickly, and that's not a good thing.