Updated

Tens of thousands of refugees looking for asylum arrived in the United States last year. But not all of them stayed. Some of them moved up to Canada, where they thought their chances of officially gaining asylum would be better.

Canadian immigration officials told FOX News that over the last three months, more than 4,000 refugees crossed into Canada from the United States seeking asylum.

Officials are seeking to slow that movement, and an agreement between the United States and Canada, called Safe Third Country (search), has changed the rules about where a person may file for asylum. The agreement went into effect Wednesday.

Canada has long been known as a country with more generous social services for refugees and less stringent asylum laws. Last week, before the new agreement took effect, one Colombian refugee fearing deportation from the United States told FOX News that crossing into Canada was his "only hope."

"We need to get there where we can have legal position to work. What we want is work," said Oscar Rodriguez, one of 400 refugees who tried to cross the Peace Bridge (search) in Buffalo, N.Y.

"We know for a fact that they have come over because they wanted to make their claim before the deadline," said Chris Kealey at the Canada Border Services Agency (search).

Immigration officials say under the new law refugees will no longer be able to use the United States as a gateway to asylum the way 13,000 refugees did last year. They first entered the United States and then traveled to Canada to claim asylum.

"It will require an individual who comes into the United States or into Canada, and is fleeing persecution, to file for asylum in the country they first landed or first enter as opposed to doing something that is informally known as asylum shopping," said Christopher Bentley of the U.S. Citizenship and Immigration Services (search), which is now part of the Department of Homeland Security.

But Gloria Rivera, a refugee advocate in Detroit, Mich., said she doesn't consider the decision of refugees to go north a matter of "shopping."

"I think people who are desperate and whose lives are in danger are going to try to get assistance, and if some of them were not successful in the United States, many of them would have a chance in Canada," Rivera said.

She added that logistics often bring asylum seekers to the United States first. More direct flights travel to the United States and more U.S. consulates are found around the world.

Rivera said she worries that the new agreement will result in more illegal border crossings.

"People, when they are desperate and they want to go to Canada, and that is their only option, they are going to find a way, and people who smuggle them are going to find them," she said.

U.S. and Canadian officials say that the Safe Third Country agreement in no way jeopardizes security at the northern border. Instead, safeguards in the agreement are designed to preserve the integrity of the system, and help those truly fleeing persecution.

Click in the box near the top of the story to watch a report by FOX News' Jeff Goldblatt.