Effective learning in neuronal networks requires the adaptation of individual synapses given their relative contribution to solving a task. However, physical neuronal systems -- whether biological or artificial -- are constrained by spatio-temporal locality. In other words, synapses can only use information available at their physical location and at the present moment in time. How such networks can perform efficient credit assignment, remains, to a large extent, an open question. In machine learning (ML), the answer is almost universally given by the error backpropagation algorithm, through both space and time (BP(TT)). However, this algorithm is well-known to rely on biologically implausible assumptions, in particular with respect to spatio-temporal (non-)locality. Alternative forward-propagation models such as real-time recurrent learning (RTRL) only partially solve the locality problem, at the cost of scaling due to prohibitive storage requirements. We introduce Generalized Latent Equilibrium (GLE), a computational framework for fully local spatio-temporal credit assignment in physical, dynamical networks of neurons. We start by defining an energy based on neuron-local mismatches, from which we derive both neuronal dynamics via stationarity and parameter dynamics via gradient descent. The resulting dynamics can be interpreted as a real-time, biologically plausible approximation of backpropagation through space and time in deep cortical networks with continuous-time neuronal dynamics and continuously active, local synaptic plasticity. In particular, GLE exploits the ability of biological neurons to phase-shift their output rate with respect to their membrane potential, which is essential in both directions of information propagation. For the forward computation, it enables the mapping of time-continuous inputs to neuronal space, effectively performing a spatio-temporal convolution. For the backward computation, it permits the temporal inversion of feedback signals, which consequently approximate the adjoint variables necessary for useful parameter updates.