In this paper, we design, analyze the convergence properties, address the implementation aspects, and numerically test the performance of AFAFed. This is a novel Asynchronous Fair Adaptive Federated learning framework for stream-oriented IoT application environments, which are featured by time-varying operating conditions, heterogeneous resource-limited devices (i.e., coworkers), non-i.i.d. local training data and unreliable communication links. The key new of AFAFed is the synergic co-design of: (i) two sets of adaptively tuned tolerance thresholds and fairness coefficients at the coworkers and central server, respectively; and, (ii) a distributed adaptive mechanism, which allows each coworker to adaptively tune own communication rate. The convergence properties of AFAFed under (possibly) non-convex loss functions is guaranteed by a set of new analytical bounds, which formally unveil the impact on the resulting AFAFed convergence rate of a number of Federated Learning (FL) parameters, like, first and second moments of the per-coworker number of consecutive model updates, data skewness, communication packet-loss probability, and maximum/minimum values of the (adaptively tuned) mixing coefficient used for model aggregation. Extensive numerical tests show that AFAFed is capable to improve test accuracy by up to 20% and reduce training time by up to 50%, compared to state-of-the-art FL schemes, even under challenging learning scenarios featured by deep Machine Learning (ML) models, data skewness, coworker heterogeneity and unreliable communication.