Undeterred by Trump, caravan buses arrive in border city

But it will likely be days before members of the group -- which organizers say is about 600 people in total -- head to the US-Mexico border to turn themselves in and ask for asylum.

They're part of a caravan that convened at Mexico's southern border weeks ago, then trekked through the country as part of an annual pilgrimage organized to bring light to the plights of migrants.

A large number of people in this year's caravan are from Honduras. Among the reasons they've given CNN for fleeing the country: widespread gang violence, domestic violence, poverty, political repression after a contested presidential election and discrimination against the transgender community.

US officials have already made it clear that they're skeptical of the migrants' motives, warning that anyone with an invalid claim will be swiftly deported and that anyone who tries to cross the border illegally will face prosecution.

Their journey spurred a tweetstorm

This year's caravan first drew President Trump's ire shortly after a Fox News segment about the group aired earlier this month.

By the end of that week, Trump had ordered National Guard troops to deploy to the border in a memo warning of a security crisis there.

While political pressure over the caravan mounted north of the border, in Mexico the migrants continued their journey.

For weeks, they've been passing through different parts of the country in a trek billed as part activist march, part humanitarian mission.

Sometimes they stowed away aboard freight trains. Sometimes they rode chartered buses. Sometimes they walked, carrying crosses and protest signs.

All along the way, they maintained they pose no threat to the country where they hope to find refuge.

"We are not bringing any guns," 32-year-old Karen Gallo told CNN from the Mexican city of Puebla.

"There are no jobs, no justice, no laws in Honduras," she said.

What will happen at the border?

It's legal for immigrants to turn themselves in to authorities and ask for asylum.

But in recent days US officials have issued a number of statements warning the approaching caravan, suggesting that even though authorities will evaluate any asylum claims once the caravan arrives, they're eyeing migrants in this group with suspicion, not welcoming them with open arms.

"These individuals -- and their smugglers -- ignored the willingness of the Mexican government to allow them to stay in Mexico," US Attorney General Jeff Sessions said in a statement Monday. "Let today's message be clear: Our nation has the most generous immigration system in the world, but this is a deliberate attempt to undermine our laws and overwhelm our system. There is no right to demand entry without justification. Smugglers and traffickers and those who lie or commit fraud will be prosecuted to the fullest extent of the law."

Those who turn themselves in and ask for asylum may end up behind bars while officials evaluate their claims "efficiently and expeditiously," Homeland Security Secretary Kirstjen Nielsen said. And those who don't have a valid claim, she said, will be swiftly deported.

Trump once again weighed in on Twitter, saying that he'd instructed his head of Homeland Security "not to let these large Caravans of people into our Country."

Rep. Zoe Lofgren, the top Democrat on the House subcommittee on immigration, accused Trump of using the caravan to stir up hatred.

"People who have a legitimate fear of persecution, under US law have a right to present their case," Lofgren told reporters Monday. "That's not a violation of immigration law. That's a part of immigration law."