We often say that arm swing should not be coached early on.

From Canton: "Relating shoulder muscle activities to upper limb kinematics suggested these muscles mainly acted eccentrically, providing evidence that passive elements are a significant factor in arm swing control. However, the conserved muscle activity patterns and temporal coupling of limb movements when pelvis motion was reduced are suggestive of an underlying active maintenance of the locomotor pattern via linked upper and lower limb neural networks."

Here are some of our deeper thoughts as to why we stand firm on this, not coaching arm swing early on.
Look at this photo, there are lots of different arm swings in every group of runners. These differences are not choices for the most part, the arms are just doing what they must, based off of many parameters in a runner, things that are working right, and not so right.
To be more clear, aberrant arm swing is often a compensation to cope with other flawed mechanics elsewhere, things such as a weak core on one side, loss of thoracic lateral bend or rotation, altered limb stability patterns, hip stability challenges etc. Thus, it is almost foolish to change an arm swing that you do not like in you or your client, because often that is not the problem. Arm swing is a power producer, but it is also a huge ballast like appendage that is used to help maintain balance changes. So, look for all possible causes of what you so, that which looked aberrant, and fix those mechanical flaws first.
From Canton: "Current research has yet to determine how passive dynamics and active neural control contribute to upper limb swing during human locomotion. The present study aimed to investigate these contributions by restricting pelvis motion during walking, thereby altering the upward energy transfer from the swinging lower limbs."
Here at The Gait Guys we have discussed for years the principles of the antiphasic nature between the pelvis "girdle" and shoulder "girdles" in that they should move in opposite rotational planes, and yet be equal in their amplitude, and that when this occurs, arm and leg swings are mostly symmetrical, equal in amplitude and symmetrical in their swing planes. This study found that when the pelvis was restricted, that the ranges of motion of the shoulder and trunk, as well as the vertical trunk center of mass movement, were also reduced, as we have said many times in our writings and in quoting the research over the years. This study also supported our long standing position that arm swing is more of a passive phenomenon, yet with complex coupling of the upper and lower limb neural networks, but also strongly taking its queues from the trunk, pelvis and leg swing.
One final thought from us, coaches, especially sprint coaches, are still going to coach arm swing and force arm swing drills, the ones they want to see, to achieve more power. . . . sigh (we get it, speed is important, but there could be a cost to making the body do what is it naturally struggling to do cleanly). So, if you are going to employ these arm swing sprint drills, get someone to fix the aberrant patterns first, if you want to see fewer injuries. Otherwise, don't be surprised if you see in your runners more thoracic lean to one side, a head tilt to one side, athletes complaining of mid or low back or neck pain, tightness, shoulder pain and the list goes on. Forcing your desired coached arm swing pattern on a clients already compensated physiology may have some unwanted costs.
-Dr. Allen (of the gait guys)

From the -Canton and MacLellan paper:
"Relating shoulder muscle activities to upper limb kinematics suggested these muscles mainly acted eccentrically, providing evidence that passive elements are a significant factor in arm swing control. However, the conserved muscle activity patterns and temporal coupling of limb movements when pelvis motion was reduced are suggestive of an underlying active maintenance of the locomotor pattern via linked upper and lower limb neural networks."
Active and passive contributions to arm swing: Implications of the restriction of pelvis motion during human locomotion.Canton S1, MacLellan MJ2. Hum Mov Sci. 2018 Feb;57:314-323. doi: 10.1016/j.humov.2017.09.009. Epub 2017 Sep 25.