  The investigation of the behaviour for geometric functionals of random fields
on manifolds has drawn recently considerable attention. In this paper, we
extend this framework by considering fluctuations over time for the level
curves of general isotropic Gaussian spherical random fields. We focus on both
long and short memory assumptions; in the former case, we show that the
fluctuations of $u$-level curves are dominated by a single component,
corresponding to a second-order chaos evaluated on a subset of the multipole
components for the random field. We prove the existence of cancellation points
where the variance is asymptotically of smaller order; these points do not
include the nodal case $u = 0$, in marked contrast with recent results on the
high-frequency behaviour of nodal lines for random eigenfunctions with no
temporal dependence. In the short memory case, we show that all chaoses
contribute in the limit, no cancellation occurs and a Central Limit Theorem can
be established by Fourth-Moment Theorems and a Breuer-Major argument.
