There's no such thing as a silly question, only a silly answer.
The rust browning process leaves a fine layer of silt-like rust on the surface of the metal. This needs removed before the next application of the solution. This is called carding.
I would venture a guess that sandpaper would clog and might effect the finish. Commonly used items for carding are coarse cloth, degreased fine steel wool, or my favorite, a carding brush. The brush has baby soft fine stainless steel bristles. I got mine from Brownells. It's also used for cleaning files.
One thing that LMF solution requires to brown properly is humidity. I use a spare closet with a small humidifier and heater. Many folks just hang the parts in their bathroom and run a hot shower for a few minutes to generate humidity. It's not a complicated process and doesn't require specific temperatures or humidity levels. Each coat takes about 2-4 hours to rust properly, depending on the humidity. 4-5 repetitions is usually enough.
There's a couple tricks I've learned. Even if the first coat doesn't look like it's rusting well, take it out in 2-4 hours, don't card it, apply another coat and let it go another 2 hours. That sets a good foundation for the next coats and lets them get a "bite" on the metal.
I'd advise not sanding the metal any finer than about 220 grit. Any finer and the solution has a hard time biting into the metal. The rust browning process hides sanding scratches well anyway, so sanding to absolute perfection isn't really needed.
Always gently wipe on the solution and never rub. Rubbing causes a copper plating color to show up and you'll need to steel wool it off or that spot won't rust. I just make one pass with the solution and don't worry too much about eveness. It always looks uneven early in the process anyway. The color evens out and darkens as you continue, and the hot/boiling water at the end beautifully darkens it to a rich, chocolate brown.
One of my favorite tricks is to let the first couple coats really rust the metal. Sometimes for 18-20 hours or more, keeping a close eye on it. This causes pitting on the metal and makes the gun look very authentic and old. You can control the pitting by the amount of time you leave it rusting. It will begin as fine frosted pitting and progress to slightly rough pitting. That's a good place to stop unless you want the barrel to look very aged. If so, allowing it to progress to deeper pitting is fine. It's very easy to control the pitting and stop it exactly when you want. You do need to do the pitting in the first coat or two though, as you will actually remove the browning if you attempt it later in the process.
To finish the process, you use baking soda in water to neutralize any solution remaining on the metal, and rinse in boiling water.