Sir Roger Gale was puzzled when a string of e-mails from Donald Trump’s presidential campaign landed in his inbox. As a Briton and a member of Parliament, Gale is barred by US law from giving Trump money, much less voting for him.

“I’ve gotten rid of most of that rubbish,’’ he said in an interview.

The e-mails to Gale were among a wave of fund-raising pleas inexplicably sent by the Trump campaign in recent days to lawmakers in the United Kingdom, Iceland, Australia, and elsewhere. The solicitations prompted watchdog groups in Washington to file two separate complaints Wednesday with the Federal Election Commission alleging that the Trump campaign was violating federal law by soliciting funds from foreign nationals.

“The scale and scope of this does seem somewhat unprecedented,’’ said Brendan Fischer, associate counsel for the Campaign Legal Center, which joined Democracy 21 in one of the complaints.

The episode is only the latest fund-raising stumble by Trump’s presidential campaign, which entered June with only $1.3 million and has been scrambling to put together a financial operation to take on the well-funded campaign of likely Democratic nominee Hillary Clinton.

Last week, Trump dispatched his first official fund-raising e-mail, casting a wide digital net for small donations that he hopes will infuse his cash-strapped campaign and the Republican Party’s coffers at a time when Clinton is lapping him in the money chase.

So far, his effort has both shown promise and hit speed bumps. His campaign said it raised $2 million in less than 12 hours after blasting out its first e-mail. But the e-mails to foreign nationals have caused a distraction, at best, and some experts have raised questions about whether his initial appeals landed in supporters’ spam folders at a higher rate than normal.

Whether the snags prove to be growing pains for a campaign that until recently eschewed traditional fund-raising or a sign of more serious stumbles to come is a key question facing Trump and the Republican Party as the general election comes into focus.

Trump’s campaign did not respond to requests for comment on the complaints to the FEC or questions about why e-mails were sent to foreign lawmakers.

The e-mails were sent both before and after Trump’s trip to Scotland last weekend to visit two of his golf courses, and their message focused on the “Brexit’’ vote in Britain to leave the European Union.

The mogul is trying to build out his once-lean campaign operation with experienced hands. As a part of that process, Trump recently enlisted the help of the Prosper Group, an Indiana-based digital strategy firm, to help with online fund-raising.

There have been other complications with Trump’s online fund-raising.

Tom Sather, senior director of research at Return Path, a data firm that performs e-mail studies, said he noticed that Trump’s campaign switched domain names when he sent his first e-mail out, causing many servers to flag it as spam and not recognize that it was coming from a familiar source.

Sather said he and his company also noticed a big jump in the size of Trump’s distribution list June 21, signaling that the campaign might have added another list or lists to its existing file.

Renting e-mail lists from former candidates is common practice in politics, and there is evidence suggesting Trump is doing that now. A Trump fund-raising e-mail sent out Wednesday afternoon came from ­info@chrischristie.com.Christie, a former presidential candidate and the Republican governor of New Jersey, supports Trump.

In last week’s fund-raising e-mail, Trump vowed to match the $2 million raised with his personal funds. Trump is also holding in-person fund-raisers across the country more often.

But he will need to keep up an intense pace if he wants to catch up to Clinton. Compared with Trump’s $1.3 million, campaign finance filings showed Clinton had $42 million at the start of this month.