I do ruler work on my domestic machine. Yes, you do move the ruler along with the quilt sandwich and it works just fine. You lower the feed dogs and use a special ruler foot and templates that are designed to be the correct thickness to prevent the ruler riding up over the template. There are a lot of other considerations when choosing a template or ruler to use when doing ruler work on a domestic sewing machine and one of the best people to ask would be Amy Johnson.
Craftsy offers a course called "Quilting with Rulers on a Home Machine" taught by Amy Johnson. She's also got a lot of information out via You tube and her blog "Freemotion quilting adventures.com" see the link in Tartan's post above.
Rob