We introduce a class of two-parameter discrete dispersion models, obtained by combining convolution with a factorial tilting operation, similar to exponential dispersion models which combine convolution and exponential tilting. The equidispersed Poisson model has a special place in this approach, whereas several overdispersed discrete distributions, such as the Neyman Type A, Pólya-Aeppli, negative binomial and Poisson-inverse Gaussian, turn out to be Poisson-Tweedie factorial dispersion models with power dispersion functions, analogous to ordinary Tweedie exponential dispersion models with power variance functions. Using the factorial cumulant generating function as tool, we introduce a dilation operation as a discrete analogue of scaling, generalizing binomial thinning. The Poisson-Tweedie factorial dispersion models are closed under dilation, which in turn leads to a Poisson-Tweedie asymptotic framework where Poisson-Tweedie models appear as dilation limits. This unifies many discrete convergence results and leads to Poisson and Hermite convergence results, similar to the law of large numbers and the central limit theorem, respectively. The dilation operator also leads to a duality transformation which in some cases transforms overdispersion into underdispersion and vice-versa. Many of the results have multivariate analogues, and in particular we consider a class of multivariate Poisson-Tweedie models, a multivariate notion of over- and underdispersion, and a multivariate zero-inflation index.