We have studied a series of Pt/Co/M epitaxial trilayers, in which Co is sandwiched between Pt and a nonmagnetic layer M (Pt, Ir, Cu, and Al). Using polar magneto-optical Kerr microscopy, we show that the field-induced domain wall speeds are strongly dependent on the nature of the top layer, increase going from M = Pt to lighter top metallic overlayers, and can reach several 100 m/s for Pt/Co/Al. The domain wall (DW) dynamics is consistent with the presence of chiral Néel walls stabilized by the interfacial Dzyaloshinskii-Moriya interaction (DMI) whose strength increases going from Pt to Al top layers. This is explained by the presence of DMI with opposite signs at the Pt/Co and Co/M interfaces, the latter increasing in strength going towards heavier atoms, possibly due to the increasing spin-orbit interaction. This work shows that in non-centrosymmetric trilayers, the domain wall dynamics can be finely tuned by engineering the DMI strength, in view of efficient devices for logic and spintronic applications.