Hey Maarten, I have used FFWinPlugin to get WebDAV working in Firefox https://msdn.microsoft.com/en-us/library/ff407576.aspx.
I changed two files:
CMSFormControls\Media\DirectUploadControl.ascx and added:
<object id="winFirefoxPlugin" type="application/x-sharepoint" width="0" height="0" style="visibility: hidden;">
CMSScripts\WebDAV.js and added:
var hownowPlugin = document.getElementById("winFirefoxPlugin");
var version = hownowPlugin.GetOfficeVersion();
hownowPlugin.EditDocument(strDocument, version);
Simply add some logic to determine the browser is Firefox and you are good to go :)
I am not aware of any other plugins to get other browsers supported but I guess the implementation will be similar.
Good luck and happy coding! :)