  A classical P\'olya urn scheme is a Markov process whose evolution is encoded
by a replacement matrix $(R_{i,j})_{1\leq i,j\leq d}$. At every discrete
time-step, we draw a ball uniformly at random, denote its colour $c$, and
replace it in the urn together with $R_{c,j}$ balls of colour $j$ (for all
$1\leq j\leq d$).
  We are interested in multi-drawing P\'olya urns, where the replacement rule
depends on the random drawing of a set of $m$ balls from the urn (with or
without replacement). This generalisation has already been studied in the
literature, in particular by Kuba & Mahmoud (ArXiv:1503.09069 and 1509.09053),
where second order asymptotic results are proved for $2$-colour urns under the
balanced and the affinity assumptions.
  The main idea of this work is to apply stochastic approximation methods to
this problem, which enables us to remove the affinity hypothesis of Kuba &
Mahmoud and generalise the result to more-than-two-colour urns. We also give
some partial results in the two-colour non-balanced case.
