We show that minimal models of Q-factorial NQC log canonical generalised pairs exist, assuming the existence of minimal models of smooth varieties. More generally, we prove that on a Q-factorial NQC log canonical generalised pair (X, B + M ) we can run an MMP with scaling of an ample divisor which terminates, assuming that it admits an NQC weak Zariski decomposition or that KX + B + M is not pseudoeffective. As a consequence, we establish several existence results for minimal models and Mori fibre spaces in dimensions at most 5. ContentsWe would like to thank Jihao Liu and Kenta Hashizume for useful discussions related to this work.2020 Mathematics Subject Classification: 14E30.