Recently, the authors showed that Reed-Muller (RM) codes achieve capacity on binary memoryless symmetric (BMS) channels with respect to bit error rate. This paper extends that work by showing that RM codes defined on non-binary fields, known as generalized RM codes, achieve capacity on sufficiently symmetric non-binary channels with respect to symbol error rate. The new proof also simplifies the previous approach (for BMS channels) in a variety of ways that may be of independent interest.