  Consider a branching random walk evolving in a macroscopic time-inhomogeneous
environment, that scales with the length $n$ of the process under study. We
compute the first two terms of the asymptotic of the maximal displacement at
time $n$. The coefficient of the first (ballistic) order is obtained as the
solution of an optimization problem, while the second term, of order $n^{1/3}$,
comes from time-inhomogeneous random walk estimates, that may be of independent
interest. This result partially answers a conjecture of Fang and Zeitouni. Same
techniques are used to obtain the asymptotic of other quantities, such as the
consistent maximal displacement.
