Let
$G$
be a connected nilpotent Lie group. Given probability-preserving
$G$
-actions
$(X_i,\Sigma _i,\mu _i,u_i)$
,
$i=0,1,\ldots ,k$
, and also polynomial maps
$\phi _i:\mathbb {R}\to G$
,
$i=1,\ldots ,k$
, we consider the trajectory of a joining
$\lambda $
of the systems
$(X_i,\Sigma _i,\mu _i,u_i)$
under the ‘off-diagonal’ flow
\[ (t,(x_0,x_1,x_2,\ldots ,x_k))\mapsto (x_0,u_1^{\phi _1(t)}x_1,u_2^{\phi _2(t)}x_2,\ldots ,u_k^{\phi _k(t)}x_k). \]
It is proved that any joining
$\lambda $
is equidistributed under this flow with respect to some limit joining
$\lambda '$
. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg’s approach to the study of multiple recurrence. It is also shown that the limit joining
$\lambda '$
is invariant under the subgroup of
$G^{k+1}$
generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.