Tijuana, Mexico (CNN) After a difficult, monthlong journey from Central America to the US-Mexico border, dozens of asylum-seeking migrants are vowing to remain outside an immigration processing center until "every last one" is admitted into the country, an organizer with the caravan said late Sunday.

Earlier, the migrants marched from Friendship Park in Tijuana, Mexico to the San Ysidro port of entry. They stood on the Mexican side; on the other side lay San Diego, California. It was the final leg for some in the caravan of hundreds of migrants, which had reached Tijuana on Tuesday.

Alex Mensing, an organizer with Pueblo Sin Fronteras, which assembled the caravan, said 50 migrants were admitted to the immigration processing center on the Mexico side. However, that is incorrect. The migrants congregated on a bridge leading to the US border while waiting to be processed by American officials.

Before the group arrived, US Customs and Border Protection officials said the port had already reached full capacity, and migrants trying to get into the United States may need to wait in Mexico as officials process those already in the facility.

Some migrants said they had walked the last leg of the journey filled with anxiety. Others scarfed down food before they filed into the center, afraid there would be no food once they turned themselves in to border officials.