Recent results suggest a pressure-induced transformation of bilayer graphene to a diamond-like structure. Intriguingly, the reported transformation is absent or incomplete in multi-layer graphene beyond two layers, implying the suppression of the transformation by an unclear mechanism. In this work, we use reactive molecular dynamics simulations to describe the pressure-induced structural evolution of multi-layer graphene with two to six layers to pressures up to 200 GPa. The results show that bilayer graphene transforms into a diamond-like film, i.e., more than 75% sp3 hybridized atoms. In contrast, three- to six-layer graphene with AB layer stacking fail to form a diamond-like film up to 200 GPa. Bond formation analysis indicates that the transformation or its suppression is related to the generation of a bond chain structure across layers, connecting sp3 hybridized atoms. The result suggests that the ABA layer stacking is directly related to the hindering of the chain structure formation and the suppression of the transformation. Surprisingly, the bond-chain-structure transformation mechanism is effective in three-layer graphene with ABC layer stacking, as well as five-layer graphene with ABCAB layer stacking. That suggests that stacking faults are effective catalysts for the high-pressure transformation of multi-layer graphene to diamond.