Okounkov bodies and restricted volumes along very general curves

Given a big divisor $D$ on a normal complex projective variety $X$, we show that the restricted volume of $D$ along a very general complete-intersection curve $C\subset X$ can be read off from the Okounkov body of $D$ with respect to an admissible flag containing $C$. From this we deduce that if two big divisors $D_1$ and $D_2$ on $X$ have the same Okounkov body with respect to every admissible flag, then $D_1$ and $D_2$ are numerically equivalent.