What happens when a new browser comes out? Do I have to wait for the GWT compiler to be updated?

Definitely not!

GWT was designed in such a way to have all browser-specific code in user-level libraries. What's more, JavaScript itself has very consistent support across browsers, so when the GWT compiler performs its Java-to-JavaScript compilation, it doesn't need to worry about which browser the JavaScript is being generated for.

That said, what makes compliance to newer browsers tricky is the DOM API. For backwards-compatible browsers, it just works. For other cases, it's straightforward to change the user-level libraries. All that's required is to implement a version of DOMImpl for the desired browser. In some cases an entirely new browser may require a bit more work to be supported by GWT, but this would be the exception rather than the rule.
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