I wasn't able to recreate the issue you're having, nor have we had any other similar reports. We haven't made any changes to Chat recently, so it's unlikely that it's a newly-introduced bug of some kind.
Clearing the cache won't hurt anything, but it would be a surprise if it it fixes anything.
How many people are experiencing this issue? Is it just Moderators viewing the queue, or are any regular users reporting any issues with connectivity?
How many different browsers/computers are having this issue? Have you tried with a different computer or browser? (See my note below about Chrome.)
Are you using Macs, PCs, or both?
Also, it's possible that you have some extensions that may be interfering (like AdBlock or FlashBlock). To check your extensions, click the little hamburger menu in the far upper right of your Firefox browser. Choose Add-Ons from the popup. On the far left menu, choose Extensions, and you'll see any extensions that you have installed. You might want to try disabling all of them to start with, and see if that makes a difference for you. Then you can start adding them back, one at a time, until you find one that causes the problem.
I know you said you were having the issue occasionally with Chrome. I am also having trouble with Chrome occasionally, where it becomes completely unresponsive, but it doesn't seem to matter what site I'm on. I can fix the problem sometimes by refreshing the page, but lots of times I have to quit and restart Chrome (my formerly preferred browser) to get it to work.