Preheat the grill using indirect or two-zone grilling targeting 350 - 375 degrees Fahrenheit. Place a medium-sized cast iron pot in the grill over indirect heat to let it warm up.
When the grill reaches target temperature add butter and mushrooms to the cast iron pot and sauté until browned (arund 8-10 minutes). Remove the mushrooms from the pot set aside. If the mushrooms are big, or the pot is small, consider browning the mushrooms in two batches.
With mushrooms removed, take the clove of garlic and rub it all over the warmed pot to give a flavor and essence of garlic to the pot. Add the wine, cover grill, and bring to a simmer (about 4 minutes).
In a small bowl, combine the cornstarch and the Kirsch and whisk to combine. Then add to the simmering wine.
Finally add the shredded cheeses to the pot (a little at a time) and stir to combine. Continue stirring fondue to incorporate the cheeses and the liquid. After about 5 minutes the cheese should be well combined, the look should be shiny, and the fondue should be thick and stretchy. You can test it using a wooden spoon. Dip the wooden spoon into the mixture. It should lightly coat the spoon (but not be too thick), and slowly drip back down into the pot.
Serve immediately in the pot, or transfer to a fondue pot with a sterno heat unit for serving.