If I'm understanding this one correctly, this isn't exactly a bug - and has nothing to do with playing via a mobile browser.
The condition you describe is also apparent on the main site: You have to refresh the page or Chat stream (e.g. by making a move or hitting Enter on the Chat window) to see anything that's been added since you last loaded the page. It's not like Chat in Live or instant messaging; it was designed for correspondence.
At some point, we'll likely redo echess with a bunch of sweet HTML5 goodness, and at that point we might see something more like real-time/dynamic updates. Until then, the only echess avenue that offers that so far is the iOS app.
When playing Online Chess (Correspondence Chess) in Google Chrome version <21.0.1180.75 against an opponent on an iPhone 4, I've noticed some strange behavior in the chat tab.
I don't get chat messages from my opponent until I send a message. Then it'll print his queue to my screen. This also works if I just click the enter button or push my enter key with no text in the input box. Also, refreshing the entire page doesn't get his queue sent to me and it doesn't fix the problem.
I said the version was less than version 21.0.1180.75 because I just updated and haven't had a chance to check that the problem remains with the updated version. I also haven't checked with any other web browsers yet, and I'm running on a MacBook Pro with OSX Lion.