For hyperbolic systems in one spatial dimension ∂tu + C∂xu = f(u), u(t, x) ∈ ℝd, we study sequences of oscillating solutions by their Young-measure limit, μ, and develop tools to study the evolution of μ directly from the Young measure, v, of the initial data. For d ≤ 2 we construct a flow mapping, St, such that μ(t) = St(v) is the unique Young-measure solution for initial value v. For d ≥ 3 we establish existence and uniqueness of Young measures that have product structure, that is the oscillations in direction of the Riemann invariants are independent. Counterexamples show that neither μ nor the marginal measures of the Riemann invariants are uniquely determined from v, except if a certain structural interaction condition for f is satisfied. We rely on ideas of transport theory and make use of the Wasserstein distance on the space of probability measures.