It almost goes without saying, but when working with tools sharp tools cut best. Router bits, like every other cutting device ever made, gets dull after prolonged use.
So if you're struggling to make clean cuts check the router bit. It may just need cleaning or it may need replacing. We recommend replacing your router bit with a carbide 1/4" down cut spiral bit.
Nothing good happens when both the material being cut and the machine making the cuts both move! Think of it this way. When using a table saw the blade is spinning and you push the material through the blade. Can you imagine what would happen if both the blade and material moved at the same time, things would get extremely dangerous!!!
To prevent this from happening we use double sided tape to secure the inlay material to a backing material. We also use tape to hold the inlay template securely in place against the material being cut. Then we clamp the entire assembly to a work bench so nothing but the router moves during the cutting process.