Abstract A general framework is presented that allows for Minimalist grammars to use arbitrary movement operations under the proviso that they are all definable by monadic second-order formulas over derivation trees. Lowering, sidewards movement, and clustering, among others, are the result of instantiating the parameters of this framework in a certain way. Even though weak generative capacity is not increased, strong generative capacity may change depending on the available movement types. Notably, TAG-style tree adjunction can be emulated by a special type of lowering movement.