Frank said the caravan's rapid growth underscores "how desperate the Honduran people are - that they'd begin walking toward refuge in the United States with only a day pack full of belongings".
The march comes days after Vice President Mike Pence met with the presidents of Honduras, Guatemala and El Salvador.
Hernandez said last month that cuts in US support for Central America would hinder efforts to stem illegal immigration, and welcomed China's growing diplomatic presence in the region as an "opportunity".
Honduras is one of a dwindling number of countries that still has formal diplomatic ties with Taiwan.
Thousands of migrants have fled Honduras and other Central American nations in an effort to escape the poverty and violence that has engulfed their homelands.
The march started with fewer than 200 members in the Honduran city of San Pedro Sula one of the world's most risky places.
The migrants, carrying backpacks and bottles of water, eventually pushed past the Guatemalan officers.
Guatemala said in a statement on Sunday that it did not promote or endorse 'irregular migration'. Guatemalan police initially blocked migrants from reaching a customs booth, Reuters images showed. It was not clear how long the standoff lasted, but the group was ultimately able to cross, said march organizer Bartolo Fuentes, a former Honduran lawmaker.
But U.S. officials - and President Trump - warned the marchers.
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