A crosslinkable elastomeric composition includes at least one elastomeric polymer including carboxylic groups, and at least one epoxidized elastomeric polymer including greater than or equal to 0.1 mol % and less than or equal to 20 mol % of epoxide groups relative to a total number of moles of monomers present in the at least one epoxidized elastomeric polymer. The composition is crosslinkable in a substantial absence of additional crosslinking agents. A tire for vehicles includes a carcass structure, a belt structure extended coaxially around the carcass structure, and a tread band extended coaxially around the belt structure. The tread band comprises an external rolling surface intended to come into contact with the ground. In one embodiment of the present invention, the tread band includes at least one crosslinked elastomeric material. A process for producing the tire is also disclosed.