In non-commutative probability, a notion of independence can be understood as a rule to compute mixed moments. For each notion of independence, the respective cumulants can be defined. Relations between different brands of cumulants have been studied recently. In this talk, we will focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants and noncrossing partitions — those formulas were missing in the literature. The approach is based on a pre-Lie algebra structure on cumulant functionals. The relations among cumulants are described in terms of the pre-Lie Magnus expansion.