As has been pointed out, there were not "orphan train routes" as such. When kids were herded onto trains and "sent west" the trains went wherever the tracks were.

Old maps will show where the railroads ran, and should be fairly easy to find. ANY ROUTE to any town you could buy a ticket for likely was at one time or another a host to an individual train. For information on when one of these trains may have gone through a given town/area, it probably will be necessary to search in the "local histories" of the area of interest.

Many towns, townships, counties, and even states have published "Histories of ... ," often to celebrate "centennials" and/or semi-centennial reminiscences. Libraries in larger towns are fairly likely to have a copy of any such published histories, and state (and occasionally local) historical societies exist in some places. The passage of one of the trains would have been a significant enough event for someone to have written a "memory piece" about it.

A surprising number of small town newspapers have survived with some archives, or the archives of ones that folded are sometimes "adopted" and preserved by successors.

Railroad workers were fairly commonly "unionized" and "union newspapers" found by accident have contained "years ago" columns. Local Rail buffs might be able to steer you to an archive. (These were most often monthly publications, so a small archive can cover a lot of years, if you can find someone who can take a look or help you to look.)

A few old newspapers quoted in local histories have indicated that prior to running a group of orphans through a given area "recruiters" sometimes passed through to "sell the idea" of adoptions, before any orphans were sent from "back East" and local old newspaper archives might report the appearance of one of these even if they didn't mention when a train went through. (Check the want ads?)

Also vaguely reported was that adoption of the orphans sometimes was by an "auction" (slave auction?) conducted by the orphans' escorts, on the pretense that funds from the auction would be returned to the orphanages back where that crop of orphans originated. I've seen no reports of any such funds being accounted for, but maybe ... some of it did get back (?). Some of it most likely just bought a ticket to the next town for the orphans that "didn't get placed," until that group ran out of orphans.

John