The real needs are inside Venezuela, where malnutrition is growing and cases of chronic diseases are spreading.
While the United States has been turning the screws on Caracas through financial sanctions and other measures, it has also said it is prepared to deliver humanitarian aid to the country.
Earlier this week, U.S. Secretary of State Rex Tillerson said the White House was considering using resources earmarked for Venezuela to help Colombia confront the migratory crisis.
That Venezuela refuses to accept aid is something of a regional mystery. Despite its own hardships, the administration often jumps at the chance to provide aid abroad. After hurricanes Maria and Irma last year, the government sent food and emergency crews to the Caribbean, and Caracas also sent aid to Ecuador after that country's 2016 earthquake.
While some argue Venezuela won't take aid out of ideological pride, others see more nefarious reasons.
Ruben Chirino Leanez, a political analyst and pollster, said the crushing shortages in the country give the administration power.
"To force people to stand in line for hours for a roll of toilet paper, a piece of bread or a bit of meat -- in a sense that's a form of crowd control," he said.
People have little energy for protesting and rallies, he suggested, "when even the most basic items are at risk."
That the scarcity is causing an exodus may be icing on the cake for the Maduro administration. "You're losing people who are probably opposed (to the administration) anyway," Chirino said.
For Colombia, the influx is a role reversal. During the country's six-decade civil war, millions of people were displaced -- including more than 1 million who fled to Venezuela.
On Thursday, Santos acknowledged the historical debt with the neighboring country, and asked Colombians to tame any "xenophobic" impulses they might have.
"Venezuela was very generous when Colombians (left) looking for a better life," he said. "We, too, have to be generous with Venezuelans."
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