Short question: What happens to the beta-binomial distribution, when n increases to infinity? Is there a count distribution arising like it's for the classical binomial distribution?
Consider the urn model for the beta-binomial:
... imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, it is replaced and another black ball is added to the urn. If this is repeated n times, then the probability of observing k red balls follows a beta-binomial distribution with parameters n,α and β.
When both $\alpha$ and $\beta$ are very large, the additional balls introduced do not appreciably change the chances of red or black. This is just a binomial experiment (with fixed probabilities $\alpha / (\alpha + \beta)$ of red and $\beta/ (\alpha + \beta)$ of black).