So a significant portion of "business things" that a workflow/business process system has to handle consist of things done in browsers.
One avenue has traditionally been IEMech (moribund at the moment due to OLE/COM complexity that has changed in later versions), but there are also different remote control solutions available for Firefox and Chrome.
Firefox's FF-Remote-Control is a great little add-on that works quite well. For the time being, therefore, Firefox is going to be my automated browser of choice even though Chrome is currently my actual browser.
For Chrome, the situation is somewhat different, as Chrome's security model doesn't permit an add-on to listen on a port. As a result, the Chromi extension hits a running server on localhost (the Chromix remote-control system). It doesn't seem as flexible as FF-Remote-Control, but I haven't spent much time with it yet.
So: for now, Firefox.