For years, thousands of U.S. and NATO troops have stood guard in Poland—a presence designed to deter an invasion by Russian troops and tanks.

This week, the Western alliance faced a less conventional challenge: At least 2,000 people from the Middle East trying to cross into Poland from Russia’s closest ally, Belarus.

European officials accuse Belarus of abetting human traffickers bringing migrants into their country and then of funneling the new arrivals toward the border in an effort to provoke a crisis—things Belarus denies.

Tensions are rising. Poland has deployed more soldiers along the frontier to keep the would-be crossers out. Russia says it views the troop movements as a threat and has responded by sending bombers to patrol over Belarus.

Alexander Lukashenko, Belarus’s authoritarian president, noted in a press conference that the Russian warplanes were capable of carrying nuclear weapons.

“What we are dealing with is a new type of war,” Polish Prime Minister Mateusz Morawiecki said in a Facebook post Thursday. “This is a war in which civilians and media messages are the ammunition.”

Poland’s government says Belarus is engaged in an “act of hybrid aggression” to provoke a clash at the borders—playing out in full view of the world. Other Western officials say they think Russian President Vladimir Putin wants to create a dramatic spectacle and undermine the West, but stop far short of actual armed conflict.

The defense ministers of Estonia, Latvia and Lithuania warned Thursday of the risk of military confrontation. Russia, which has denied any involvement, said the European Union should speak with Belarus to resolve the crisis.

Belarusian Defense Minister Viktor Khrenin said its neighbors’ military activities—particularly in Poland—are unrelated to the migrant crisis and may signal they “are ready to unleash a conflict in which they want to involve Europe in solving their internal political problems, as well as problems related to relations within the European Union.”

The standoff involves far fewer migrants than the flows that poured into the continent after 2015. On most days, only a few dozen asylum seekers are managing to get across the border. But the imagery of cold and hungry migrants, directed by Belarusian troops toward a single border checkpoint, has played out on social media for days, roiling European politics.

Belarus on Thursday allowed staff from the United Nations’ refugee agency and International Organization for Migration to visit a makeshift camp near the border housing roughly 2,000 people, including children and women, many pregnant, the agency said Friday. Aid workers are racing to provide assistance as winter approaches.

“The makeshift camp at the border with no adequate shelter, food, water and medical care in freezing temperatures is not a safe and suitable place for people and could lead to further loss of life,” the U.N. agency said in a statement.

Several refuge-seekers have died at the border since the crisis began this summer. Belarus’s border agency said Friday that a migrant teenager was treated for hypothermia overnight by Belarusian doctors and later returned to his family.

Since summer, Belarus has issued tourist visas to people from Iraq, Syria, Yemen and other countries in what EU officials say is an orchestrated effort to get them to the EU’s borders.

European officials have been pressing airlines across the Middle East and Turkey to significantly limit the number of passengers they carry to Minsk, winning some promises of cooperation.

Belarus and Russia are unlikely to risk a military clash against the North Atlantic Treaty Organization, officials and analysts say. Instead, Moscow and its allies are seeking to sow dissension. In recent years, tactics have ranged from disinformation to holding back gas supplies while energy prices in Europe soar. At a government meeting Thursday, Mr. Lukashenko floated the idea of halting gas transit from Russia to the EU. The Kremlin said Friday that wouldn’t happen.

For Europe, the big fear is a repeat of 2015, when more than a million asylum seekers crossed into the continent, creating scenes that fed a surge in popularity for nationalist parties and helped nudge the U.K. out of the EU. The numbers camped in Belarus now are comparably small: a few thousand.

“This is as much a PR campaign as a breach-of-the-border campaign,” said Elisabeth Braw, a senior fellow at the American Enterprise Institute, a Washington-based think tank. “They know how divisive immigration is and how these images will create a toxic divide between citizens and different countries’ governments.”