Could the police still take them back home against their will or otherwise prosecute them?
Once the search goes out...
Once the police get the report of a missing person and they actually do file it (which can be tricky if the person is in the 18-21 gap but much easier below 18), the missing person is in their system. Contrary to popular belief, there is no 24 hour waiting period to report a minor missing, that is a rule that only regards adults and is not universal for all agencies.
Once the file is in the system, the report is reported upwards, because the states are forced to report any missing people under 21 to the NCIC under the Crime Control Act of 1990 - 42 U.S.C § 5779a (transferred to 34 U.S.C § 41307). This means, even if they are 18 now, the initial report to the NCIC still went up, possibly back when it was initially reported, maybe even still if the report happens after the 18th birthday. And they are still a missing child in the eyes of the NCIC.
Each Federal, State, and local law enforcement agency shall report each case of a missing child under the age of 21 reported to such agency to the National Crime Information Center of the Department of Justice.
The entry in the NCIC database however will remain even long after the 21st birthday, as it can only be removed by the initial reporting agency:
Missing Person: Adult and Juvenile: Retained until cancelled[sic!] by entering agency.
The database for missing children is for example used by missingkids.org and can be searched. Using a search for people disappeared before their 18th birthday and now aged above 50 resulted in 373 cases [+ the test-file] that are listed there. The oldest in that sample is Melvin Horst, who disappeared in 1928 and would be 96 by now. And yet, his entry is still online and can only be removed by the originally reporting agency. At the same time, more than 500 cases are people that disappeared below 18 and are now 18-21, about 400 of which disappeared between their 16th and 18th birthday.
A similar database is the Charley Project, which sources from the same database and the initial reports - and their list has more than 13000 cases. It's not separated between kids and adults. They too list Melvin Horst, their 7th oldest case, their youngest case is from fall 2019 as they only list cases older than a year due to practical limitations.
How each state handles found children or young adults that are reported missing is dependant on the state. Often, the act of the child running away on the own volition (or the report the child gives to police once apprehended) will trigger an inquiry by the CPS (or equivalent) into the living situation and could mandate foster care for some time or give some reasons for full emancipation - however that would be very dependant on the individual case.