. . . dominance of the lumbosacral girdle over the cervicothoracic is probably preserved in humans
This suggests that arm swing is, to a notable degree, subservient to leg swing.
Research thus far has strongly suggested two pieces to arm swing, a passive and an active swing component. Without muscle activity, passive swing amplitude and relative phase decrease significantly. As phase decreases, it is referred to as in-phase swing pattern of the arms. The Goudriaan et al paper referenced below concluded that "muscle activity is needed to increase arm swing amplitude and modify relative phase during human walking to obtain an out-phase movement relative to the legs."
But it is more complicated that this . . . .
Research continues to suggest that interlimb coordination is achieved at the brainstem and cortical level, which this study suggests as to why we can dual task and walk with something in our hands, carry objects and even walk and run with said objects and changes in our gait . . . . because, the program is running off a top down neurologic mediated process with predictable, economically CPGs(central pattern generator) in place.
"The coordination of arm and leg movements takes the form of an in-phase relationship between diagonal limbs . 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."-Preece et al.
Human Movement Science 45 (2016) 110–118
The coordinated movement of the spine and pelvis during
Stephen J. Preece, Duncan Mason, Christopher Bramah
School of Health Sciences, University of Salford, Salford, Manchester M6 6PU, United Kingdom
Gait Posture. 2014 Jun;40(2):321-6. doi: 10.1016/j.gaitpost.2014.04.204. Epub 2014 May 6.
Arm swing in human walking: what is their drive? Goudriaan M1, Jonkers I2, van Dieen JH3, Bruijn SM4.