In this study, we examine the frequency and physical drivers of transformations from cool-core (CC) clusters into non-cool-core (NCC) ones, and vice versa, in a sample of 352 massive galaxy clusters (M500c = 1014-15.3 M ) from the TNG-Cluster magnetohydrodynamical cosmological simulation of galaxies. By identifying transformations based on the evolution of central entropy and focusing on z . 2.5, we find that clusters frequently undergo such events, depending on their diverse assembly and supermassive black hole (SMBH) histories. On average, clusters experience two to three transformations. Transformations can occur in both directions and can be temporary, but the ones to higher-entropy cores, i.e., in the qualitative direction from CC to NCC states, make up the overwhelming majority. The CC phases are also shorter than NCC phases, and thus overall the TNG-Cluster population forms with low-entropy cores and moves toward NCC states as cosmic time progresses. We study the role that mergers play in driving transformations, and find that mergers within $∼$1 Gyr prior to a transformation toward higher- (but not lower-) entropy cores occur statistically more often than in a random control sample. Most importantly, we find examples of mergers associated with CC disruption regardless of their mass ratio or angular momentum. However, past merger activity is not a good predictor for z = 0 CC status, at least based on core entropy, even though clusters undergoing more major and minor mergers eventually have the highest core entropy values at z = 0. We therefore consider the interplay between AGN feedback and evolving cluster core thermodynamics. We find that core transformations are accompanied by an increase in AGN activity, whereby frequent and repeated (kinetic) energy injections from the central SMBHs can produce a collective, long-term impact on central entropy, ultimately heating cluster cores. Whether such fast-paced periods of AGN activity are triggered by mergers is plausible, but not necessary.