The dielelctric breakdown model (DBM) is generalized to include lower cutoffs, which prevent growth at low fields. The new models may represent realistic situations in some DBM and some viscous fingering experiments. Multifractal theory is shown to provide quantitative predictions for the crossover from the usual DBM patterns (at small finger sizes) to a new, spiky, behavior (at large sizes). For one of the models, the theory also predicts when growth will stop. The predicted crossover scaling function is confirmed by numerical simulations.