I think if you search "grip refinishing" you'll find some detailed discussions. here's a cliff notes version of what I do:
1) Soak in a small, covered pan of acetone for a day or two to dissolve and remove oils and prior finish
2) Allow to dry 1 day
3) Hand-rub a coat of tung oil (from any hardware store - satin or gloss, your choice) on - a THIN, LIGHT coat, but rubbed vigorously into the wood. Latex/nitrile gloves help keep it off your hands; it stains and clings. This also helps avoid leaving finger prints in the finish.
4) Allow to dry for one day- no direct heat. In a window or a few feet away from a heater is ok but no direct heat. On top of the fireplace hearth in winter is ideal.
5) repeat 3) and 4) at least three and as possibly more times.
Three coats will give a nice durable finish. Six or more can develop into a gorgeous glass-like finish. Either way, I only coat the underside of the grips once or twice - that's enough for protection, the rest is for wear and shine, which don't matter on the underside. Some folks advocate doing a sanding down with 1000grit before the final coat - depends upon how you want the final coat to look. I just make sure the wood finish is sanded to the level I want before finishing.
Eagle grips are usually made from an oily Indian rosewood. The acetone will take on a dark brown/reddish color quickly from that wood type and its natural oils leaching out. You may need to change the acetone bath once when using Indian rosewood. The wood may come out looking blanched, pale and awful. Don't worry. The tung oil will bring it back to it's natural color and luster.
If a prior urethane coat was applied it can curl up and be clingy in the bath. Just brush it off with a toothbrush as it loosens. Eagle doesn't use that so that would have been someone else's garage finish.
Here's a pair of Eagle grips which I refinished for just this reason - someone else's poor application of a urethane or acrylic finish that started flaking. When done, they were gorgeous: