Poisson distribution is commonly used to model count data with infinite support. But,overdispersion, mainly due to excess of zeroes or high values, undermines model performances. Poisson mixtures offer an elegant and appealing modelling strategy to overcome such limitations. However, the underlying mixing distribution need to be carefully selected. In this work, we focus on how the tail behaviour of the mixing distribution is related to the tail of the resulting Poisson mixture. We define five classes of mixing distributions and we identify for each case whenever the Poisson mixture is in, close to or far from a domain of attraction of maxima. We also characterise how the Poisson mixture behaves similarly to a standard Poisson distribution when the mixing distribution has a finite support. Finally, we study, both analytically and numerically, how goodness-of-fit can be assessed with the inspection of tail behaviour.