A very light dusting of baby powder (or something close to that), then brushed thoroughly out, will do the trick. Initially it lightens up the hair but when you put it in, but it returns to its normal shade as you brush it out and the powder takes off the shine in the process.

But baby powder has a very distinctive perfumy smell to it that I don't care to carry around with me, so I looked through the drugstore and settled on Gold Bond "ultimate body powder', basically baby powder for adults. The aroma was very mild compared to regular baby powder and it was cheap. You might even find some powder that's unscented that will do the same thing.

I have a backup synthetic long hair system that I slap on when I don't have time to deal with tape and glue. Synthetic is shiny by nature but the powder makes it look very close to my human hair system. Takes just a tiny bit. Shake it onto a brush, shake out the brush, and then comb it through. It washes right out next time you rinse the system, unlike jojoba oil which may take shampoo to remove.

An added bonus is that if you use tape in front like I do, the powder gets through the lace and cuts the shine on the tape at the same time as the hair.