We study quantum transport properties in quasiperiodic magnetic superlattices by examining the motion of electrons in Fibonacci and Thue-Morse sequences. It is found that transmission resonances exhibit complicated oscillations and selective suppression. The number of resonant peaks decreases and their distribution becomes less regular in comparison to those of the periodic magnetic superlattice. It is confirmed that both doublet and singlet resonances exist in the quasiperiodic structure. It is also confirmed that complete resonant tunnelling can occur in quasiperiodic systems due to the existence of completely transparent eigenstates. The role played by a local magnetic barrier in the magnetic superlattice is investigated. The results indicate that both the transmission coefficient and the conductance are drastically suppressed with increment of the strength of a local barrier.