This is a refinish so there is already finish in the checkering. Every stock I have ever refinished (and I am well past a hundred) with checkering I have stripped then recut/cleaned out the checkering. Applied the stock finish two coats at a time reducing it back to the wood with #0000 steel wool, this fills the wood pores with the finish. This pore filling process takes time. During this process you "do not" put finish in the checking, you only keep it clean of steel wool dust with a soft tooth brush. I am an oil finish person and the oil whether it has dries or not I apply a drop or two at a time with my index finger tip, your not slopping it on with a paint brush so it's not hard to control a drop or two and keep it out of the checkering. During the application of the final finish coat I will sparingly put a couple drops in each checkering panel and spread it well about with a soft tooth brush, this will soak in well enough to freeze the checkering and protect the wood and give it the same colour tone as the rest of the stock. This is not rocket science if it needs a little more finish it can be applied a drop or so at a time with a finger tip. The checkering can be touched up with the checkering tools if need be as the finish literally will put a new light on the points and lines and you may see imperfections that until now went un-noticed.
You may notice many new factory produced guns, a Winchester M-70 featherweight with the fleur de le pattern comes to mind, the checkering is a lighter duller colour. That's because the whole stock has it's finish sprayed on then the checkering is machine cut/lazer through the finish afterwards. This way Winchester doesn't have to keep a room full of checkering Elves. The Elves did better work, trust me. Winchesters machine cut checkering is a pure nightmare to recut.
On a new fresh wood stock I will partially fill or completely fill the pores depending on the piece of wood, then put a light coat on the surface. Then I will lay out the checkering panels and cut them all in until I am satisfied that all the line depths and diamonds are even. This takes a lot of time and the stock is handled a lot so the finish protects the wood from oily, dirty hand prints.
This is generally the way I have done it for the last 40 yrs. or so. I am self taught, and I have had very good results and no complaints from anyone, including me and I am my worst critic.
Hope this helps and gives you some ideas.