Within-class normality refers to the observed variables, not the latent ones.
In the non-mixture case for continuous-normal outcomes, variances and covariances (and means) are sufficient statistics and are therefore the only quantities that the model needs to fit. In the non-mixture case for count outcomes, there are no such sufficient statistics so the model needs to be fitted to raw data.
In the mixture case, there are never any such sufficient statistics but the model needs to be fitted to raw data.
and this paper: Kreuter, F., Muthén, B., 2008. Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling, Journal of Quantitative Criminology.
Yes, GMMs also assume normality within class for the outcomes. It's true that this happens when both the growth factors and the residuals are normal within class - so there isn't much of a difference here. With numerical integration with respect to continuous latent variables such as growth factors, e.g. with categorical or count outcomes, the assumption of normal latent variables is explicitly made.
Mplus also allows within-class non-normality that Mplus achieves for skew-normal and skew-t.
If I understand correctly when the observed indicators are specified as counts (nb; negative binomial) in the GMM, the latent growth parameters are still explicitly assumed as normal.
How does changing the specification of indicators from continuous to counts (nb) change the estimation of growth parameter variation? Is it just that growth parameters are now estimated with an appropriate link?