TIJUANA, Mexico—Fermín Campos arrived at the El Chaparral border crossing to the U.S. with 11 family members six weeks ago, well before a caravan of more than 5,000 Central Americans arrived in this city across the border from San Diego.

On a notebook ledger organized and maintained by asylum seekers here to take orderly turns entering the U.S. to make their claims, Mr. Campos, a Mexican citizen, wrote the names of all the members of his large family and began waiting for his number to be called.