Just to note that firefox uses the -MOZ- prefix and Safari/Chrome use the -WEBKIT- prefix. So if a website uses a style command prefixed with -webkit- and not -moz-, Mozilla Firefox won't be able to see it the same.
So, what is the solution... My website is not displaying the updates to it unless I switch to the Internet Explorer Browser. I have been using the Firefox Browser with no website display issues utilizing another webhost.
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After submitting a support ticket...
The issue is usually related to caching; on first view, the browser saves a local copy for faster loading later. If the browser doesn't check if the page has been updated, it doesn't download a new version to display, and just shows the old one. Switching browsers works because each browser stores its cache differently.
For testing purposes if you need to forcefully clear the cache, firefox/IE usually use the control+f5 method - f5 alone is refresh, but holding control also tells it to ignore the cache.