"over the border" is correct and usable in written English.
You can use it to describe something that is crossing a physical or metaphorical boundary: "Determined to escape their oppressive regime, the refugees fled over the border to a free nation."
Africa analysts speculated anxiously about the potential for a civil war, and Fitzgerald said that a conflict in Guinea might "spill over the borders and reinfect Sierra Leone and Liberia".
Mr. Clinton acknowledged that increased American aid to counter-narcotics operations in Colombia, if successful, was likely to "cause the problem to spill over the borders" into neighboring countries like Brazil, Ecuador, Peru and Venezuela.
Should H7N9 spill over the borders of China to nearby countries such as Vietnam (through local, unregulated poultry trade), or more distantly by bird flyways to Africa, it is likely to cause an even greater number of human cases.
To get around this, some have shifted production over the borders.
Still, negotiations over the borders seem to have achieved some progress.
Some escaped Saddam Hussein's brutal rule more recently by slipping over the borders.
Politicians talk constantly of Europe being "under siege", of millions streaming over the borders.
With 5,000 Liberian civilians dead and 400,000 refugees streaming over the borders, neighboring states feared chaos.
In order to avoid a new gallery shape over the borders of the piston's cross section, several constraints need to be satisfied in the optimization.
Thousands of captured ethnic-Albanian men were this week freed from prison and shoved, dazed and battered, over the borders into Albania and Macedonia.
"The worst kind of predictions about what could happen internally, and spilling over the borders of Syria, are certainly within the realm of the possible now," she said.
