More than a million Venezuelan migrants have entered Colombia in the past 15 months, according to official estimates, and more than 4,000 have been arriving at Ecuador's border every day.
Many have been walking or hitching rides for weeks and are exhausted by the time they reach the frontier.
Early on Saturday, about 300 Venezuelans were lined up at the Rumichaca border crossing outside the Colombian city of Ipiales, and many said they had no passports to gain entry to Ecuador.
Gabriel Malavolta, a 50-year-old mechanic, left Venezuela three days ago aiming to make the overland route to Lima, Peru, through Ecuador.
He has a passport but his fiancee, Yenny, had only an ID card.
"I don't know what we're going to do, but we can't go back. I'm not sending my fiancee to go back and go hungry," he told Reuters news agency at a Red Cross tent.
"You've no idea what it's like [in Venezuela]. Whole families eat from the trash."
Another migrant, Regulo Guaita, said: "That was truly a surprise. We found out today. It's very sad because there are many, many Venezuelans leaving and (with this measure) they won't let them leave. I don't know what they'll do now."
With the flow of Venezuelan migrants causing tensions across the region, Peru's government announced immigration measures similar to Ecuador's on Friday. Passport requirements for Venezuelans will begin on 25 August.
Brazil, which neighbours Venezuela, has also expressed concerns and temporarily closed the border earlier this month. Violence has flared in the border state of Roraima where thousands of Venezuelans live in precarious accommodation.
In the border town of Pacaraima on Saturday, several migrant encampments were attacked by angry residents.
The attacks followed a protest in the town after a local restaurant owner was allegedly robbed and beaten up by Venezuelans.