A sandpile model with dynamically varying critical heights is studied in one dimension. The model displays self-organized critical behavior with an avalanche exponent τ=1.35±0.02, which correspond to the value found from a mean-field argument. The time evolution of the mass and height of the pile is characterized by a time-correlation function with a crossover time that increases systematically with system time. The average transit time of a grain through the model is proportional to the system size, but the distribution of transit times has a power-law tail. A modified model, which includes non-local interactions from force propagation within the pile, belongs to the same universality class, although the distribution of transit times changes significantly.