Hi Hugo, my history is similar to this.
I started in print design in '99. I started learning web design in '01, and gradually transitioned completely into web design by '03. The jobs I had during that transition allowed me to learn web design while I was the Chief Designer for print, because we had a lack of human resources.
With a background in graphic design, I had a natural tendency toward the technical aspects of web design. I was hand-coding HTML, CSS, PHP, JS etc by '03. This technical knowledge helped me get i the door for more technical design roles. My big break was when I took a job as the Creative Director at a UX agency. I learned so much about UX there, from first-hand experience. I wasn't thinking about it at the time, but this was the perfect way to transition from design to UX.
Now, my CV showed that I had worked at a UX agency. By '05 I was taking IX and UXD contracts based on that experience.
There's no formal 'bridge' between the two, that I know of. As always, learn by doing, and pick a way to get yourself in the UX sphere of work. Just by being in the industry, you'll learn a lot and gain credibility in that way.
Early on for me, I had similar experiences. I'd been a Creative Director for many years, but my years in UX were drawled by that. It's hard going from a senior / directorial role to a lower role, and most employers don't like to do that. The way I got around it was to focus on UX jobs that had some crossover (e.g. heavily design-focused) and word my CV so that my UX experience was highlighted.