  Given $\mu$ and $\nu$, probability measures on $\mathbb R^d$ in convex order,
a Bass martingale is arguably the most natural martingale starting with law
$\mu$ and finishing with law $\nu$. Indeed, this martingale is obtained by
stretching a reference Brownian motion so as to meet the data $\mu,\nu$. Unless
$\mu$ is a Dirac, the existence of a Bass martingale is a delicate subject,
since for instance the reference Brownian motion must be allowed to have a
non-trivial initial distribution $\alpha$, not known in advance. Thus the key
to obtaining the Bass martingale, theoretically as well as practically, lies in
finding $\alpha$.
  In \cite{BaSchTsch23} it has been shown that $\alpha$ is determined as the
minimizer of the so-called Bass functional. In the present paper we propose to
minimize this functional by following its gradient flow, or more precisely, the
gradient flow of its $L^2$-lift. In our main result we show that this gradient
flow converges in norm to a minimizer of the Bass functional, and when $d=1$ we
further establish that convergence is exponentially fast.
