Electronic/magnetic transitions and their structural consequences in Fe-based Mott insulators in a regime of very high static density are the main issue of this short review paper. The paper focuses on the above-mentioned topics based primarily on our previous and ongoing experimental HP studies employing: (i) diamond anvil cells, (ii) synchrotron X-ray diffraction, (iii) 57 FeMössbauer spectroscopy, (iv) electrical resistance and (v) X-ray absorption spectroscopy. It is shown that applying pressure to such strongly correlated systems leads to a number of changes; including quenching of the orbital moment, quenching of Jahn-Teller distortion, spin crossover, inter-valence charge transfer, insulator-metal transition, moment collapse and volume collapse. These changes may occur simultaneously or sequentially over a range of pressures. Any of these may be accompanied by or be a consequence of a structural phase transition; namely, a change in crystal symmetry. Analyzing this rich variety of phenomena we show the main scenarios which such strongly correlated systems may undergo on the way to a correlation breakdown (Mott transition