We construct a hermitian metric on the classifying spaces of graded-polarized mixed Hodge structures and prove analogs of the strong distance estimate [6] between an admissible period map and the approximating nilpotent orbit. We also consider the asymptotic behavior of the biextension metric introduced by Hain [12], analogs of the norm estimates of [19] and the asymptotics of the naive limit Hodge filtration considered in [21].
