You can't really go right in response to headers/lifts - it's too late at that point (see below). Taking advantage of those gradual course-wide shifts is a game of anticipation. If we were scientific enough to measure and chart the wind before a race, we would likely see some steady patterns to work with...since this isn't usually practical, use the forecast to see which way the wind is expected to shift to throughout the day.

This is something that Astella's online sailing simulator (can't recall the name at the moment) helped me to really visualize and I offer you this exercise. Visualize a race course in front of you with A-mark near the ceiling. Hold your hands out as the "boats" who are dead even on port tack with one further left and one further right. Now, give them both an equal header (the wind is shifting right across the entire course) and turn your hands appropriately as the skippers compensate for the header. Notice now that the boat on the left immediately lost ground on the boat to the right and is now looking more at his stern whereas they were dead even before the shift. It's an instant boost to the boat that was on the side the wind shifted to (upwind).

If you wait for the header before you go right it's too late. You have to be there when it happens and the more cross course separation you have from your opponent(s), the more gain you make if the shift shows up. Playing the anticipated course shift is a little gamble but if it's gradual and persistent, it's usually worth a boat length or two if you position yourself properly for it - but you only gain on boats that were on the wrong side for the shift.