We consider curves γ: [0 , 1] → R³ endowed with an adapted orthonormal frame r: [0 , 1] → SO(3) . We wish to deform such framed curves (γ, r) while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of r^′r^T), and a global constraint prescribing the initial and terminal values of γ and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ, r) that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.