  In this paper, we study the tangent space to a sub-Finsler manifold in the
measured Gromov-Hausdorff sense. We prove that the metric tangent is described
by the nilpotent approximation, generalizing the sub-Riemannian result.
Additionally, we study the blow-up of a measure on a sub-Finsler manifold. We
identify the new notion of bounded measure which ensures that, in the limit,
the blow-up is a scalar multiple of the Lebesgue measure. Our results have
applications in the study of the Lott-Sturm-Villani curvature-dimension
condition in sub-Finsler manifolds. In particular, we show the failure of the
CD condition in equiregular sub-Finsler manifolds with growth vector (2,3),
equipped with a bounded measure.
