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(CNN) —  

A month after beginning their arduous trip north from Central America, a caravan of migrants is so close to their goal of reaching the United States they can see the lights of its shops.

Around 100 migrants are camped out on the Mexican side of a border crossing here, waiting for a chance to enter.

It’s cold – around 55 degrees Fahrenheit – and men, women and children are lying shivering on concrete as rain begins to fall, wondering whether they should give up for the night.

To their right, people have been coming and going freely through the turnstiles that lead to the US – its billowing Stars and Stripes flag visible. But the migrants’ future is unknown.

They have traveled on foot, by bus and by train, carrying their possessions – and sometimes their children – and hope of a better life.

The caravan is both a humanitarian and an activist mission, as organizers created the event to draw greater attention to the migrants’ plight.

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

The migrants are congregating on a bridge leading to the US border while waiting to be processed by American officials. A US Customs and Border Protection spokesman says they do not have the capacity to process asylum claims at the moment and it’s not clear when they’ll have the space to accept them. They’ll have to wait on the Mexican side until their claims can be heard.

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Migrants wait at the US-Mexico border to have their asylum claims processed.

Read: Migrants vow to wait at border until they are granted asylum

Waiting game

Overnight Sunday, around 20 adults and children are sleeping on the walkway to the port of entry.

Some of the migrants are sick and you can hear them coughing as they try to sleep.

Gabriela Hernandez and her two children, Omar, 6, and Jonathan, 2, huddle under blankets near the turnstiles. It eventually got too cold and they moved to a more protected area.

A legal volunteer loans the children a phone so that they could watch cartoons in Spanish, and some other kids are playing with straws that volunteers brought them from McDonald’s and eating sugar.

Volunteers have been bringing them coffee, milk, sugar and chocolates and, earlier, Mexican officials let them out of the center to get pizza for the children.

They’re resolute. They’re not going anywhere.

But they are aware of the uncertainty of their fate. There is no welcome mat rolled out here. US President Donald Trump has vowed to do what he can to prevent the caravan of migrants to enter.

Before this group arrived, CBP 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.

Until now, Gabriela was hopeful – they all were – but, as it’s getting later and later into the night, she voices uncertainty.

“I don’t know what to think anymore,” she tells me.

It’s unclear what Monday holds – it’s an hour-by-hour situation – and for now she plans to hunker down.

Fellow migrant Tesla Rich says she remains hopeful: “Of course I have faith. It’s going to be fine.”

CNN’s Susannah Cullinane, Eric Levenson and Tal Kopan contributed to this report.