We investigate the evolution of thick points under two natural dynamics for the Gaussian free field (GFF) in dimension 2. The first dynamic we analyze is the Ornstein-Uhlenbeck GFF. We prove that, simultaneously for all points, the evolution of their thickness is continuous. Additionally, we characterize all deterministic functions $f: \mathbb{R}\rightarrow \mathbb{R}$ such that there are points whose thickness function is $f$. The second dynamic we study is the stationary solution of the additive stochastic heat equation. In this case, the thickness of points is not continuous. Moreover, this rougher dynamic generates super-thick points, namely points with thickness greater than $2$. As a function of $γ> 2$, we identify infinitely many phase transitions corresponding to the existence of exceptional times where at least $N$ points are $γ$-thick. These phase transitions, occurring at $γ^2 = 8, 6, 16/3, \dots$, converge to $4$ as $N \to \infty$. Mapping to the critical FK-model parameter via $q=4\cos^2(4π/γ^2)$, these critical values correspond to the Beraha numbers, which are precisely the points at which the CFT for critical FK percolation should be minimal