Exciton states in type-II InP/InGaP and GaSb/GaAs self-assembled quantum dots and quantum-dot superlattices subject to a normal magnetic field are calculated. Strain is explicitly taken into account in single particle models of the electronic structure, while an exact diagonalization approach is adopted to compute the exciton states. Strain reverts type II band alignment in InP quantum dots to type I, therefore no transitions between the lowest energy states of different angular momenta are observed. On the other hand, strain increases the barrier for the electron in the conduction band of GaSb/GaAs quantum dots, therefore the exciton, being composed of electron and hole states of various angular momenta, may have a finite angular momentum in the ground state. Consequently, the oscillator strength in the InP single quantum dot and quantum-dot superlattice increases with the magnetic field, while the angular momentum transitions between the bright and the dark exciton states in the GaSb system bring about decay of the oscillator strength when the magnetic field exceeds a certain value.