Let $\mathcal{M}_n(E)$ denote the set of vectors of the first $n$ moments of probability measures on $E\subset\mathbb{R}$ with existing moments. The investigation of such moment spaces in high dimension has found considerable interest in the recent literature. For instance, it has been shown that a uniformly distributed moment sequence in $\mathcal M_n([0,1])$ converges in the large $n$ limit to the moment sequence of the arcsine distribution. In this article we provide a unifying viewpoint by identifying classes of more general distributions on $\mathcal{M}_n(E)$ for $E=[a,b],\,E=\mathbb{R}_+$ and $E=\mathbb{R}$, respectively, and discuss universality problems within these classes. In particular, we demonstrate that the moment sequence of the arcsine distribution is not universal for $E$ being a compact interval. On the other hand, on the moment spaces $\mathcal{M}_n(\mathbb{R}_+)$ and $\mathcal{M}_n(\mathbb{R})$ the random moment sequences governed by our distributions exhibit for $n\to\infty$ a universal behaviour: The first $k$ moments of such a random vector converge almost surely to the first $k$ moments of the Marchenko-Pastur distribution (half line) and Wigner's semi-circle distribution (real line). Moreover, the fluctuations around the limit sequences are Gaussian. We also obtain moderate and large deviations principles and discuss relations of our findings with free probability.