Toss the cheeses and cornstarch in a medium bowl; set aside.
Rub the garlic all over the inside of a fondue pot, then discard. Add the wine and lemon juice to the pot and bring to a simmer over medium-high heat. Add the cheese mixture in batches, whisking until melted before adding more. (The mixture may look separated or curdled at this point.)
Reduce the heat to medium and add the sherry, mustard and cayenne pepper; continue cooking, whisking, until incorporated, 1 more minute. Keep warm over low heat. Serve with bread.