On the topic of central pattern generators (CPGs) and gait / locomotion
"If quadrupedal coordination is deeply embedded in the human nervous system then one might expect this to be revealed in conditions when there is a conflict between voluntary arm movements and walking. For example, Muzii et al.  combined a walking and a clapping task at preferred rates. Hand clapping was found to be tightly coupled to heel strike. When instructed to walk and clap at different rates (e.g. walk normally but clap faster) the subjects were not able to perform this task, implying that the walking rhythm dominated the coordination. Hence coupling is fairly robust, a finding that was confirmed by the observation that the typical 1:1 diagonal coordination during gait is maintained even when either one of the limbs involved is loaded with an extra 2 kg." - P. Meyns et al. / Gait & Posture 38 (2013) 555–562
We have discussed this same thing during our "dual tasking" blog posts. These things can be learned and modified with attentive training, but is it strongly suggested that the underlying CPG patterns are fairly robust. This is not to say that leg swing is the king, that it runs the show, but it seems dominant. And as the Meyns paper reviews, there is an influence from the upper limbs in terms of enhancing and shaping the overall movement and coordination of all 4 limbs.
And as the Meyns paper states, "although the connections go both ways, it is clear in the to date animal models studied, that "the caudorostral connections seem to be the most powerful ones." Meaning, the pelvis and lower limb motor patterns and pattern generators seem to dominate over the upper limbs and upper pattern generator centers.
"The dominance of the lumbosacral girdle over the cervicothoracic is probably preserved in humans as well. For example, Sakamoto et al.  showed that during combined arm and leg cycling, the cadence of the arms was significantly altered when leg cycling cadence was changed. The opposite, however, was not true, i.e. the arms did not affect the leg cadence." Meyns et al.
And, "the authors concluded that ‘‘the neural signal induced by the upper limb movements contributes not merely to enhance, but to shape the lower limb locomotive motor output, possibly through interlimb neural pathways’’.-Myens et al.
The how and why of arm swing during human walking
Pieter Meyns a,1, Sjoerd M. Bruijn a,b,1, Jacques Duysens a,c,*
P. Meyns et al. / Gait & Posture 38 (2013) 555–562