Moving Day!
Changing your line of play to stay in the pocket sounds like a good plan when the lanes change, but it may not be as simple as it sounds.
What's the problem?
As balls roll through the oil, it moves. Some of the balls absorb it, some drag it down toward the pins, and some do a little of both. In any case, pretty soon there's a track of dry through the lane heads where there was oil when you started. And there's a track of oil on the back ends where there was none earlier. Depending on who's bowling with you, the tracks are small and all in one place, or wide and scattered all over. What to do depends on how you bowl and how the lanes break down.
What kind of bowler are you?
If you're a true stroker, you're probably comfortable pretty far outside on the lanes, even out toward the first arrow. If a true cranker, you probably start near or inside of the third arrow. The part in between these two areas is for the (makes sense, doesn't it?) tweeners.
What do you do when it happens?
Stroker: As your line dries out, you may be able to make a 2 & 1 move (2 with your feet, 1 with your target) or two toward the center of the lane, to chase the oil line. You can't move much, or you'll run into the tweeners' dried out area. You might be able to just increase ball speed to stay on line, maybe move a bit outside with your feet to overcome the lack of hook in the (now oily) back ends.
Cranker: You can just keep moving inside with feet and target, until you have to loft the ball over the gutter cap (works better on the side away from the ball return, naturally). You can always increase speed, but too much of that and you run the risk of over-skid in the carry-down. The same way the strokers' inside move is limited, you can't move outside very much to defeat the carry-down, or you run into the tweeners' dry heads.
Tweener: You have a bigger problem. At first you can chase the oil line inside, maybe once or twice. But soon, if there are crankers inside you, you'll have to make a big move (called "bump, bump, and jump"). If you don't make this big jump and commit to it, you'll be right where the crankers started (but they had to leave, remember?). You could speed up and inch back outside, but if the strokers are moving in from there now, same problem. Of course, if there are no strokers or no crankers bowling with you, your options open up in that direction. If you have both strokers and crankers on your pair, you may need to become one or the other for the rest of the night.
Of course, any of you could stay where you are and change balls. Generally, that means a ball with a weaker coverstock to get through the dry heads, and a strong core or layout to defeat the carry-down oil in the back ends. Many pearlized balls are designed just for this purpose.