  We consider diploid bi-parental analogues of Cannings models: in a population
of fixed size $N$ the next generation is composed of $V_{i,j}$ offspring from
parents $i$ and $j$, where $V=(V_{i,j})_{1\le i\neq j \le N}$ is a (jointly)
exchangeable (symmetric) array. Every individual carries two chromosome copies,
each of which is inherited from one of its parents. We obtain general
conditions, formulated in terms of the vector of the total number of offspring
to each individual, for the convergence of the properly scaled ancestral
process for an $n$-sample of genes towards a ($\Xi$-)coalescent. This
complements M\"ohle and Sagitov's (2001) result for the haploid case and
sharpens the profile of M\"ohle and Sagitov's (2003) study of the diploid case,
which focused on fixed couples, where each row of $V$ has at most one non-zero
entry.
  We apply the convergence result to several examples, in particular to two
diploid variations of Schweinsberg's (2003) model, leading to Beta-coalescents
with two-fold and with four-fold mergers, respectively.
