The sweeping tariffs on Canadian, Mexican and Chinese products that President Trump imposed on Tuesday could strain the system that collects import duties and the government agencies that enforce those fees, trade and legal experts said.

Collecting import duties is usually a routine task, but the new tariffs are being imposed on Mexican and Canadian goods, many of which have been imported into the United States duty-free for many years. Adding to the challenge is the sheer volume of goods subject to the new tariffs — U.S. imports from China, Mexico and Canada totaled over $1.3 trillion last year, or about two-fifths of all imports.

The tariffs apply a 25 percent duty on goods from Mexico and Canada and an additional 10 percent on imports from China.

Importers typically employ customs brokers to calculate and pay tariffs to the government agency that collects them, U.S. Customs and Border Protection.

Adam Lewis, a co-founder and the president of Clearit, a customs broker, said that it would not be hard to tweak software to collect the new tariffs, but that a crucial part of the tariffs payment system might need significant adjustments. Importers must buy a “customs bond,” a type of insurance that guarantees the duties will be paid. Mr. Lewis said some customers might have to increase the size of their bonds to cover the extra tariff payments.

“Many of their products were coming in duty-free, and all of a sudden there’s going to be a 25 percent increase,” he said. “It’s quite large.”

In addition, policing importers for tariff evasion will now become a much bigger task for Customs and Border Protection and the Department of Justice. Some importers may try to avoid tariffs by understating the cost of goods in customs declarations or by falsely claiming they were imported from countries not subject to tariffs.

“The greater the breadth and severity of these new tariffs, the greater the likelihood that at least some potential importers may want to misrepresent the value or the origin of their goods,” said Kirti Vaidya Reddy, a former federal prosecutor who is now a partner at the law firm Quarles.

If the government finds that an importer has not paid duties, customs officials are likely to demand that the importer pay what is owed and a penalty that can double or even triple the amount due.

In a statement, a customs agency spokeswoman said: “The dynamic nature of our mission, along with evolving threats and challenges, requires C.B.P. to remain flexible and adapt quickly while ensuring seamless operations and mission resilience. These tariffs will help maintain America’s global competitiveness and protect American industries from unfair trade practices.”

Some evasion cases have become the subject of criminal prosecutions. Last year, a Miami importer pleaded guilty to participating in an import scheme involving Chinese truck tires that the Justice Department said had cost the United States more than $1.9 million in forgone tariff revenue.

But stepping up enforcement efforts is likely to require that the Justice Department devote significantly more staff to pursuing tariff evasion cases, which, lawyers said, can take time to build.

“The Department of Justice has the personnel and infrastructure to do it, but these cases are complex, transnational and document-heavy,” said Artie McConnell, a former federal prosecutor who is a partner at the law firm BakerHostetler. “You can’t rush it, and prosecutions likely won’t come quickly.”