As mountain glaciers are retreating, they are becoming increasingly debris-covered. We want to better understand how these glaciers respond to a changing climate. Here, we present an improved model that simulates transport of debris within and on the glacier. We find that short-term changes have a low impact due to long response times. However, long-term warming can lead to fast collapse of the glacier tongue after a phase of thinning, where debris cover expands and thickens.