Teen climate activist Greta Thunberg was joined by swelling and excited crowds of American teenagers at a protest outside the UN headquarters in New York on Friday, in a further blossoming of the youth environment movement given extra thrust by the Swede’s transatlantic boat crossing.

Some US children said they were at their first ever climate demonstration; others said they had been passionate about the environment for a while but had been galvanized to act by Thunberg’s rising profile.

On Friday afternoon, Thunberg and two young activists were spontaneously invited inside the UN for a meeting with a senior leader, described as “very supportive”.

Just two days after Thunberg disembarked from a yacht in New York, following two weeks on rough seas crossing from the UK, young protesters dominated the crowd of up to 1,000 outside the United Nations skyscraper in Manhattan.

They came together to demand politicians and older generations take urgent and comprehensive action to reverse the climate crisis.

Carrying hand-drawn placards with messages such as “United behind the science” and “Act now or we will”, children and young people of all ages surged into a park in front of the flags of the world outside the UN on Friday morning.

Thunberg sat cheerfully but pensively in the middle of the rally, which had a rather more earnest than festive atmosphere. Young speakers gave spontaneous speeches or led chants of “System change, not climate change” and “Don’t just watch us, join us”.

Alexandria Villaseñor, 14, who has been protesting every Friday outside the UN since December, was in her usual spot and said she had been inspired by Thunberg’s school strike campaign.

She said: “Greta being here will really galvanise students just because of how much of an inspiration she is. Everyone who’s been striking on Friday was really empowered by Greta and the action she was taking.”

Thunberg held her trademark “skolstrejk för klimatet” (Swedish for “school strike for climate”) sign, which she was seen carrying from the racing yacht on Wednesday, after refusing to take a flight to the US because of the polluting emissions.

Greta Thunberg outside the UN headquarters in New York, on 30 August. Photograph: Alba Vigaray/EPA

For Dana Henao, 16, from Brentwood, Long Island, Friday marked her first climate protest.

“The government isn’t taking enough action to protect the environment and all they care about is the money they make with corporations polluting the planet and I think we should put a stop to it,” she said. “The young people are the only ones taking action and we want to call attention to this.”

She added: “She [Greta] is really popular and she’s, like, the face of the movement.”

On Friday afternoon, Thunberg, Villaseñor and Xiye Bastida Patrick, 17, a Fridays for Future organiser, went inside the UN for an impromptu meeting with general assembly president María Fernanda Espinosa after receiving an invitation during the protest.

The trio were given a tour – including a preview of the general assembly podium, at which Thunberg will speak during next month’s climate action summit, and had a meeting with Espinosa, where they spoke about the summit, global heating, forest fires around the world, the Amazon fire and single-use plastics.

Villaseñor said the president was “very supportive”.

She said: “I said I want to see world leaders take climate action to reduce global greenhouse emissions and stay in line with the Paris agreement and also September 20 we will continue pushing and demanding that they do.”

Thunberg told the Guardian in an interview shortly after disembarking from the special “zero emissions” yacht that she wanted “a concrete plan, not just nice words” from leaders about taking urgent and comprehensive action to head off the climate crisis.

Villaseñor, who is thought to be America’s first school striker to join the movement and has been exchanging tips with Thunberg online, had been among the group of American climate activists who welcomed her to the US when her yacht docked at North Cove Marina.

She said: “What’s really important about Greta being here today is it’s the start of something new. Because with the climate summit coming up it’s the way for all the youth to unite here and send a message to world leaders at that climate summit. So even though Greta’s voyage on Malizia ended a couple of days ago, the climate action we will take on this continent has really just begun.”

Alexandria Villaseñor participates in a youth climate change protest in front of the UN headquarters in New York, on 30 August. Photograph: Jeenah Moon/Reuters

On Wednesday, Thunberg said her generation is having to “clean up” the climate crisis. She also told Donald Trump, who denies there is a climate crisis, to “listen to the science”.

On Friday, Almy Batis, 21, a student at Arizona State University, said she saw Friday as a landmark moment for America’s climate movement.

“This is a huge moment that Greta is finally here and there are Fridays for Future events scattered across the US. Her movement is going to be amplified in North America … and grow a lot.”

Esme Ruiz, eight, from New York, attending with her father, said: “The world is literally melting right in front of our eyes and no-one even realises it. We need to teach them..It’s hard to make changes sometimes, but if you need to save the world, I’d do it rather than die.”

Meanwhile, a Brooklyn teacher said she has seen a “Greta effect” in her school.