In this paper, we present a distributed reinforcement learning strategy for morphology-independent life-long gait learning for modular robots. All modules run identical controllers that locally and independently optimize their action selection based on the robot's velocity as a global, shared reward signal. We evaluate the strategy experimentally mainly on simulated, but also on physical, modular robots. We find that the strategy: (i) for six of seven configurations (3-12 modules) converge in 96% of the trials to the best known action-based gaits within 15 min, on average, (ii) can be transferred to physical robots with a comparable performance, (iii) can be applied to learn simple gait control tables for both M-TRAN and ATRON robots, (iv) enables an 8-module robot to adapt to faults and changes in its morphology, and (v) can learn gaits for up to 60 module robots but a divergence effect becomes substantial from 20-30 modules. These experiments demonstrate the advantages of a distributed learning strategy for modular robots, such as simplicity in implementation, low resource requirements, morphology independence, reconfigurability, and fault tolerance.