  Consider the supercritical branching random walk on the real line in the
boundary case and the associated Gibbs measure $\nu_{n,\beta}$ on the
$n^\text{th}$ generation, which is also the polymer measure on a disordered
tree with inverse temperature $\beta$. The convergence of the partition
function $W_{n,\beta}$, after rescaling, towards a nontrivial limit has been
proved by A\"{\i}d\'ekon and Shi in the critical case $\beta = 1$ and by
Madaule when $\beta >1$. We study here the near-critical case, where $\beta_n
\to 1$, and prove the convergence of $W_{n,\beta_n}$, after rescaling, towards
a constant multiple of the limit of the derivative martingale. Moreover,
trajectories of particles chosen according to the Gibbs measure $\nu_{n,\beta}$
have been studied by Madaule in the critical case, with convergence towards the
Brownian meander, and by Chen, Madaule and Mallein in the strong disorder
regime, with convergence towards the normalized Brownian excursion. We prove
here the convergence for trajectories of particles chosen according to the
near-critical Gibbs measure and display continuous families of processes from
the meander to the excursion or to the Brownian motion.
