This is a really difficult hypothesis to test because of the numerous confounders. Just arguing from incentives: once the US criminalizes some drug, it obviously has an incentive to make other countries follow suit, but...
The US efforts will be proportional to how much the criminalized drug comes from abroad. A recent example would be China and fentanyl.
Beijing first agreed to make all variations of fentanyl a controlled substance during trade talks between Trump and Xi in early December on the sidelines of the G20 summit in Argentina.
The primary purpose of the meeting was to avoid further escalations in the raging trade war between Washington and Beijing, which has seen billions of dollars of tariffs placed on US and Chinese goods.
The announcement of a starting date for the new laws comes amid ongoing trade talks between the two countries, which have raised hopes of a potential deal to lift tariffs and ease tensions. Chinese officials declined to link the two issues at Monday's press conference.
If there's substantial opposition inside the US for the criminalization of some drug, e.g. marijuana, then the US will do little to try to impose its views elsewhere, for example in the case of Canada or Uruguay. Even more so if US imports of said drug are negligible in contrast to its domestic production.
Countries such as the United States have historically wielded their political influence and power to encourage full implementation of the drug treaties. However, with Uruguay’s law entering its fifth year since passage, there has not been a concerted U.S. government effort to punish Uruguay bilaterally or in an international arena, suggesting that Uruguay’s reforms will not be stymied because of international pressures. In this regard, Uruguay has taken advantage of felicitous timing, with its law’s passage having come in the midst of a major shift toward cannabis regulation within the United States. After the November 2012 ballot initiatives to legalize cannabis in the states of Colorado and Washington, U.S. President Barack Obama’s administration adopted a policy of conditional accommodation of state-level cannabis legalization, contained in Justice Department enforcement guidance known as the “Cole Memo.” This accommodation provided Uruguay a political cushion internationally, just as the Uruguayan parliament was preparing to approve the country’s cannabis reform.
In the wake of the Colorado and Washington ballot initiatives, the U.S. federal government was suddenly in an awkward spot. The United States was the key architect and for decades the chief enforcer of the UN drug treaties, including vigorous enforcement of the global prohibition on non-medical uses of cannabis. To oppose Uruguay’s new law or even pressure Uruguay to revise or annul it—as it is easy to imagine previous administrations attempting to do so—would open the United States to charges of hypocrisy.
That should also hint you that there are memetic political culture forces at play worldwide. If some country argues a drug is bad for its citizens, there will be politicians in other countries that will argue the same (either because of scientific data or just because of some "moral standards") regardless of external political pressure. But it's really hard to disentangle coercion from copying. Note that unlike fentanyl, China banned opium all by itself. Also note, that in the EU at least, most NPS (new psychoactive substances) banning laws have resorted to explicit harms-related criteria for their bans, since coming up with an exhaustive list of drugs turned out to be on the losing side of the battle:
But at least some of these countries also have an agency entrusted with drawing up a list based on those principles (as a practical shortcut):
The US also has had for quite a while a Federal Analogue Act. But there's a lot less worldwide harmonization in this area (NPS), list-wise, than on the (fewer) drugs explicitly listed by the UN conventions. The Wikipedia article on the UN Single Convention has a good historical background how countries were divided according to their incentives during the negotiations. The US was among the group of countries
Having no cultural affinity for organic drug use and being faced with the effects that drug abuse was having on their citizens, they advocated very stringent controls on the production of organic raw materials and on illicit trafficking. As the principal manufacturers of synthetic psychotropics, and backed by a determined industry lobby, they forcefully opposed undue restrictions on medical research or the production and distribution of manufactured drugs. They favored strong supranational control bodies as long as they continued to exercise de facto control over such bodies. According to W.B. McAllister's Drug Diplomacy in the Twentieth Century, their strategy was essentially to "shift as much of the regulatory burden as possible to the raw-material-producing states while retaining as much of their own freedom as possible."
In contrast, just 10 years later:
The conference convened on 11 January 1971. Nations split into two rival factions, based on their interests. According to a Senate of Canada report, "One group included mostly developed nations with powerful pharmaceutical industries and active psychotropics markets . . . The other group consisted of developing states...with few psychotropic manufacturing facilities".[6] The organic drugmaking states that had suffered economically from the Single Convention's restrictions on cannabis, coca, and opium, fought for tough regulations on synthetic drugs. The synthetic drug-producing states opposed those restrictions. Ultimately, the developing states' lobbying power was no match for the powerful pharmaceutical industry's, and the international regulations that emerged at the conference's close on 21 February were considerably weaker than those of the Single Convention.
Some the hardline-ness of the US policy depended, rather predictably, on whether they made the drugs in question or not. Keep in mind that the conventions separate drugs in schedules, with various degrees of control and enforcement.
