Hundreds of families arrive every day only to find they can't make the crossing due to Moreno's decision on Monday to require visas of Venezuelans who want to come into Ecuador, which has already admitted roughly a million people fleeing economic hardship and political unrest in Venezuela.
"Please, President Lenin ... Open your heart! Here there are sick children, pregnant women, elderly people, people with disabilities!," Marbella Hernandez said, addressing the Ecuadorian head of state via Efe news.
The more than 1,000 Venezuelans here are to share that message with journalists, police commanders, civil servants and just about anyone else who sets foot on the bridge, a span of 200m (656ft) across the Carchi River.
On Monday, Venezuelan migrants effectively shut down the bridge for more than eight hours, chanting "We want refuge, we want to pass" at Ecuadorian border guards unable to accede to their pleas without a change in the orders they receive from Quito.
At the front of the throng were young mothers with babies in their arms and grandmothers holding their grandchildren by the hands, a reflection of the fact that many of the Venezuelans waiting on the Rumichaca are trying to join family members already in Ecuador.
Senior Colombian police officers told Efe news that they are saddened and frustrated by the standoff.
And immigration officials on Ecuador's side of the bridge are also unhappy as they look across the bridge to see mothers and children huddled in tents provided by international aid organisations.
"We have come because of the need in Venezuela. We're cold here, but at least we're eating cookies," Odalis Mago, who is accompanied by her husband, children and several grandchildren, tells Efe news.
Mago's family has relatives living in Chone, Ecuador, but they arrived at the border crossing after the visa requirement took effect at midnight Monday.
Colombian authorities have installed some of the Venezuelans in shelters in Ipiales, the nearest city to the Rumichaca crossing.
But many of the Venezuelans insist on staying at the bridge on the slim chance that Ecuador will relent and let them in.
One of those sharing a large Red Cross tent with Mago's family is Oran Jeldona, who vows to remain at Rumichaca until Ecuador opens its doors.
"We are going to pass come what may," he says, declining to rule out an attempt to reach Ecuador through one of the score of unguarded passes in the surrounding mountains.
On arrival, each migrant receives a rudimentary supply kit and a blanket to ward off the night-time chill in this spot more than 3,000m (9,836ft) above sea-level, where the temperature dipped to 5 C (41 F) in the wee hours of Tuesday.
( With inputs from IANS )