The extinction process of fragmented populations, characterized by a small number of conspecifics inhabiting each patch, is heavily affected by natural and human disturbance. To evaluate the risk of extinction we consider a network of identical patches connected by passive or active dispersal and hosting a finite, discrete number of individuals. We discuss three types of disturbance affecting the metapopulation: permanent loss of habitat patches, erosion of existing patches, and random catastrophes that wipe out the entire population of a patch. Starting from an infinite-dimensional Markov model that fully accounts for demographic stochasticity, we reduce it to finite dimension via moment closure with negative-binomial approximation. The compact models obtained in this way account for the dynamics of the fraction of empty patches, the average number of individuals in occupied patches, and the variance of their distribution. After comparing the performance of these compact models with that of the infinite-dimensional model in the case of no disturbances, we then proceed to computing persistence-extinction boundaries as bifurcation lines of the compact models in the space of demographic and disturbance parameters. We consider bifurcations with respect to demographic and environmental parameters and contrast our results with those of previous theories. We find out that environmental catastrophes increase the risk of extinction for both frequent and infrequent dispersers, while the random loss of patches has a much larger influence on frequent dispersers. This influence can be counterbalanced by active dispersal. Local erosion of habitat fragments has a larger influence on infrequent than on frequent dispersers. We finally discuss the important synergistic effects of disturbances acting simultaneously.