  We investigate moment sequences of probability measures on
$E\subset\mathbb{R}$ under constraints of certain moments being fixed. This
corresponds to studying sections of $n$-th moment spaces, i.e. the spaces of
moment sequences of order $n$. By equipping these sections with the uniform or
more general probability distributions, we manage to give for large $n$ precise
results on the (probabilistic) barycenters of moment space sections and the
fluctuations of random moments around these barycenters. The measures
associated to the barycenters belong to the Bernstein-Szeg\H{o} class and show
strong universal behavior. We prove Gaussian fluctuations and moderate and
large deviations principles. Furthermore, we demonstrate how fixing moments by
a constraint leads to breaking the connection between random moments and random
matrices.
