Competition between unitary dynamics that scramble quantum information non-locally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. We introduce large-N Brownian hybrid circuits acting on clusters of qubits, which provide an analytically tractable model for measurement-induced entanglement phases and criticality. Such quantum circuit dynamics either partially preserves or destroys an initial entanglement shared between a set of system and reference qubits, depending on the measurement rate. Our approach can access a variety of entropic observables, which are represented as a replica path integral with twisted boundary conditions, whereas the measurement induced entanglement transition appears as a second-order phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. The transition is mean-field-like and we characterize the critical properties near the transition in terms of a simple Ising field theory in 0+1 dimensions. By coupling the large-N clusters on a lattice, we also extend these solvable models to study the effects of power-law long-range couplings on measurement-induced phases. In one dimension, the long-range coupling is relevant for α<3/2, with α being the power-law exponent, leading to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for α<1 the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases. The volume-law phase for α<1 realizes a novel dynamically generated quantum error correcting code whose code distance scales as L^(2−2α).