We report on 207Pb, 79Br, 14N, 1H, 13C and 2H NMR experiments for studying the local order and dynamics in hybrid perovskite lattices. 207Pb NMR experiments conducted at room temperature on a series of MAPbX3 compounds (MA = CH3NH3+; X = Cl, Br and I) showed that the isotropic 207Pb NMR shift is strongly dependent on the nature of the halogen ions. Therefore 207Pb NMR appears to be a very promising tool for the characterisation of local order in mixed halogen hybrid perovskites. 207Pb NMR on MAPbBr2I served as a proof of concept. Proton, 13C and 14N NMR experiments confirmed the results previously reported in the literature. Low temperature deuterium NMR measurements, down to 25 K, were carried out to investigate the structural phase transitions of MAPbBr3. Spectral lineshapes allow following the successive phase transitions of MAPbBr3. Finally, quadrupolar NMR lineshapes recorded in the orthorhombic phase were compared with simulated spectra, using DFT calculated electric field gradients (EFG). Computed data do not take into account any temperature effect. Thus, the discrepancy between the calculated and experimental EFG evidences the fact that MA cations are still subject to significant dynamics, even at 25 K.