In this work the asymptotic behaviour of the solution of the evolution problem of three‐dimensional linearized elastic curved rod‐like bodies with respect to the small thickness ε of the rod is analyzed. It is found that, when ε tends to zero, the solution converges, in an appropriate sense, to a single space‐variable function being an inextensible displacement. Together with an additional function, describing the rotation angle of the cross‐sections, it is identified as the unique solution of the one‐dimensional evolution problem of curved rods, posed on the middle curve of the rod. The differential equations of the model are obtained.