Place the grated cheeses in a large bowl. Toss them with the cornstarch, dry mustard, garlic powder, and white pepper.
Pour the wine into the fondue pot. Turn the temperature to medium-high and bring the wine to a strong simmer (bubbling, but not boiling strongly). This should take 1-2 minutes.
Stirring constantly with a wooden skewer in a zigzag (not circular) motion, add the cheeses, a handful at a time, waiting for each handful to melt completely before adding another.
Reduce the temperature to medium-low and serve the fondue with your choice of dippers. My favorite dippers are lightly steamed vegetables.
Video
Notes
I shred the cheeses coarsely using a box grater. Pre-shredded cheese is not recommended, as the anti-caking agents added to it can interfere with melting.
It's best to shred the cheeses 30 minutes ahead of time and let them slightly dry up. This will make it easier to coat them evenly with the cornstarch and spices.
If the fondue seems stringy, stir in a teaspoon of fresh lemon juice. You can also add it as a preventative when you add the wine to the pot.
Stirring the cheese in zigzag motions rather than in circles helps prevent the cheese from becoming stringy.
An electric fondue pot is the easiest because it enables you to easily control the temperature settings when melting the cheese and also when keeping it warm after it's melted. I own two fondue sets - one by Cuisinart and one by Artestia. Both work equally well.
The wine is needed here, not just for its flavor but because its acidity helps prevent the cheese from becoming stringy. You could try using light beer, milk, or chicken stock. But then it's a good idea to add a tablespoon of lemon juice to the mixture.
You can keep the leftovers in the fridge, covered, for up to 4 days. Reheat them in the fondue pot on the lowest setting (you might need to add a bit of wine), or use them cold as a cheese spread. I don't recommend trying to reheat the leftovers in the microwave, as they tend to separate, even when reheated very slowly. I also don't recommend freezing the leftovers.
The nutrition info is for the fondue only and excludes dippers.