A crosslinkable elastomeric composition includes an elastomeric polymer including epoxide groups and an oligomer of a fatty acid. The composition is crosslinkable substantially in an absence of additional crosslinking agents. Tyres including one or more components made of crosslinked elastomeric material are also disclosed. Additionally disclosed is a process for producing a tyre for vehicle wheels including manufacturing a tyre comprising at least one crosslinkable elastomeric material, subjecting the tyre to moulding in a cavity defined in a vulcanization mould, and crosslinking the at least one elastomeric material by heating the tyre to a predetermined temperature and for a predetermined time. Further, a crosslinked elastomeric product obtained by crosslinking the composition is disclosed.