We study the properties of anomalous diffusion on finite intervals. The process studied due to the presence of trapping events and long jumps is described by a double-fractional (time and space) Fokker–Planck equation. The properties of the overall process are affected not only by long waiting times and long jumps but also by boundaries. Special attention is given to the examination of the survival probability and the first-passage-time density. Using analytical arguments and numerical methods, we show that the asymptotic form of the survival probability is determined by the trapping process. For a special choice of parameters, we compare numerical results with theoretical formulae, demonstrating that numerical solutions constructed by subordination methods reconstruct known analytical results very well. Finally, we show that the power-law distribution of waiting times is responsible for the divergence of the mean first-passage time even for a power-law distribution of jump lengths.