First, how are you defining "email client"? To me an email client is a mail program installed on my computer, but it is not a web based email service such as Gmail that you mentioned in your question.
Most email clients support IMAP, in fact the more challenging question to answer these days would be to start searching for and listing email programs that still do not support IMAP.
Creating a mail filter or mail rule that forwards an email and then deletes the original copy is also not difficult with most email programs today. If you are including the free mail services such as Gmail then yes you are likely to run into more variety in what is and is not supported. Many free email services will not let you forward email. Gmail and FastMail.FM are examples of services that allow filters and forwarding.