We study phase transitions in dynamical processes on branching hierarchical lattices. The phase transitions on such lattices are strongly influenced by the structure of the lattice. This is demonstrated for two kinds of processes: an avalanche transmission process and a synchronisation process. Predictors for the two kinds of transitions, viz. continuous and discontinuous phase transitions can also be constructed by identifying the microtransitions which occur before the phase transition. Here, we identify the existence of microtransitions in the process of avalanche transmission on load bearing hierarchical networks. It has been seen in earlier work that typical realizations of this network exhibit a second order, continuous transition, whereas the critical realization of the network shows a first order, discontinuous transition. Microtransitions can be used to predict the transition point for both cases. We also study the transition to synchronisation in branching hierarchical lattices using the evolution of Chat´e-Manneville maps placed on the lattice. The maps are diffusively coupled, and the map parameters increase hierarchically, depending on the map parameters at the sites they are coupled to in the previous layer. The system shows a transition to synchronisation which is second order in nature, with associated critical exponents. However, the critical realisation of this lattice shows a transition to synchronisation which is discontinuous. This transition can thus be said to belong to the class of explosive synchronisation with the explosive nature depending on the nature of the substrate. We carry out finite-time–finite-size scaling for the continuous transition and analyze the scaling of the jump size for the discontinuous case. We discuss the implications of our results.