A man deserted from the Armory Square Hospital in DC in Feb 1863. His next recorded presence was in 1865 Iowa when he got married under an assumed name. I am curious to know how he might have made it west. Trains seem unlikely. Walking? Wouldn’t a twenty something guy walking have raised an eyebrow in 1863?
This question seems to be predicated on two assumptions:
a twentysomething guy walking through town would be suspicious. I'm not sure that is completely true. I don't know what the migration patterns for that era are, (I know the data exists, but I can't find it), I believe there were still a lot of people going West. The deserter could take any normal form of transportation (horse, coach, etc.). If challenged, the deserter could merely state that they were joining their family in the West.
An unstated assumption that anyone who was of an age for military service should be in the military, rather than heading West. I think this is where @Semaphore's point is critical; Northern troops were largely volunteers, not conscripts. The first problem with this assumption is that conscription was not yet the law at the time of the desertion "The Enrollment Act, 12 Stat. 731, enacted March 3, 1863" Wikipedia.
Exemptions from the draft could be bought for $300 or by finding a substitute draftee. history.com
The policy of substitutions was continued throughout the war. The problem with substitution was that it provided substitutes with powerful incentives to desert soon after enlisting. Career "jumpers" made a living off of enlisting as a substitute, collecting their compensation, deserting before their units were dispatched to the front, and repeating the process. The problem was well known to the military commanders who regularly saw the same recruits repeatedly. In addition, troops furnished by substitution were considered to be of an inferior quality in comparison to regulars and volunteers. Wikipedia
Maybe Mhw has seen too many thrillers set in fascist or communist states, where police officials go through trains demanding to see the papers of all the passengers. Or maybe he imagines that police would ask to see travelers' identity papers at the borders between the various counties and states. That didn't happen in 1860s America.
Remember that American society was not nearly as automated and mechanized as today, and only a minority of healthy men could be spared from ordinary jobs to serve in the army, and many of them would sometimes be traveling for work reasons, looking for new jobs, or moving west. One of the main advantages of the Union over the Confederacy was its more mechanized agriculture that enabled a lot of farm workers to join the army without lowering food productions.
A lot of men who didn't want to be conscripted headed out west to the frontier. One army officer sent west after the war found his region full of confederate sympathizers, draft dodgers, deserters from the Union army, and other sinister characters.
The US Army had a provost marshal's bureau dealing with military law and order, including tracking down deserters. But that didn't stop a small number of "bounty jumpers" from enlisting for the enlistment bounties paid by the federal, state, and local governments to encourage voluntary enlistment, deserting as soon as they could, and reenlisting in another unit for another bounty, over and over again. Even though the maximum possible penalty for desertion was death, the percentage of deserters caught was small enough that many thousands of soldiers deserted once, and there were even those "bounty jumpers" who deserted over and over again.