That reminds me of Google's GWT, which compiled Java to JavaScript and supported a limited subset of the JDK. Maintaining a legacy GWT codebase is not a very pleasant experience, because you are stuck in a Java-like abstraction layer on top of the actual browser features and the DOM.
I guess compiling a language like Rust to WebAssembly solves these issues for a part?
GWT is very different because it directly 'transpiles' Java to a high-level Javascript, you basically get the worst of both worlds ;) asm.js and WebAssembly are much lower level, they are generated from LLVM bitcode, theoretically any language that can be compiled through LLVM can be used with asm.js / WebAssembly (as long as the language runtime libraries are ported to run in a browser environment).
I don't know, but if Web Assembly allows something like dynamic linking, it would enable, for example, Oracle to supply a WASM-JVM via CDN, that JVM based languages could target.
But this would have to be a rather cut-down version of the JVM, I think, so probably not that viable.
I guess compiling a language like Rust to WebAssembly solves these issues for a part?