MP spoke in memory of her murdered colleague Jo Cox as she joined thousands of people protesting against US president Trump

His hair attracted quite a few hundred slogans in London, and many placards displayed a personal slant on his most infamous sayings about women as tens of thousands of marchers added their voices to a roar of protest on the first day of Donald Trump’s presidency. The roar was echoed in cities across the globe.

The Women’s March, a global day of action, took place in 15 British and Irish cities – and 161 around the world – and was billed as an inclusive event to stand up for equality and dignity for all.

In the capital, an estimated 80,000-100,000 people joined a cheerful gathering in the winter sunshine, with chants of “build bridges not walls”.

Women and men of all ages, children, babies and quite a few pet dogs took part in the march from the American embassy in Grosvenor Square to Trafalgar Square, where there were speeches from TV comedian and Women’s Equality party joint founder Sandi Toksvig and Labour MP Yvette Cooper.

London’s mayor, Sadiq Khan, was there, as were comedian Sharon Horgan, actor Gillian Anderson and artist Grayson Perry. “We shall overcomb”, “Free Melania” and “Keep your tiny hands off our rights” joined slogans of “Love not hate” and “Toddlers for resistance” on banners.

Holding a sign saying: “Only 3 years, 11 months and 29 days to go,” were Sue Henderson from Royal Leamington Spa, and her sister, Geraldine Cullinan. “I’m here fighting for a more respectful world. We’re not going to let Trump have things his way,” said Henderson. “I had a new year’s resolution that instead of sitting indoors clicking on things, I would get off my arse and do something, so here I am.”

Cullinan added: “It’s a relief to be here, to see that there are other people in the world who think things are not right. It feels good.”

Around a fifth of the crowd were men. Asked why he had come, David Barclay, from Reading, marching with his two young sons and his girlfriend, said: “It didn’t occur to me not to come. Equality for women is equality for everyone. My kids won’t have a better world under leaders like Trump.”

Under a placard reading “Misters for Sisters”, George and Paula Coleridge had come with friends from west London. “He’s a narcissist, Trump. He doesn’t like criticism, so it’s important to show him criticism.”

Having their picture taken with Canadian Arrested Development actor Will Arnett, a fellow protester, were the Webster family from London. Harriet, 11, said: “I want to protest for women’s rights. I’m against people like Trump who are racist, sexist, homophobic and so much more.”

Her sister Emily, eight, added: “I’m here because of the wall. It’s not very nice to the Mexicans to build a wall.”

Their mother, Lucy, said she was glad they had come, despite the cold. “I brought them because it’s a really lovely, fun march for them to be part of. Just to see that there are many people who feel like us is very inspiring.”

As she sent off the selfie with Arnett to her husband on her mobile, she said: “My husband is in LA working and he’s going on the march there.”

The numbers were too high for all the marchers to get into Trafalgar Square; many remained in the street, stretching back to the Florence Nightgale statue in Waterloo Place, where there were dozens of spirited conversations among friends and strangers.

“Its truly amazing the atmosphere. This is not the sort of thing I’d ever do but maybe I’ll just do it a bit more,” said Lucinda McLaverty from west London.

During her speech, Cooper referred to the late MP Jo Cox, whose photograph appeared on several placards at the rally. “We are marching because the most powerful man in the [US] thinks it’s OK to grab women ‘by the pussy’,” she said.

“We are marching because a talented woman MP was murdered by a far-right extremist and we need to call it out as the terrorism it is. And we are not just marching – we’re … standing up to the misogynists, the bullies and the haters who try to intimidate and silence people online, just as for years they tried to intimidate or silence women on the street.

“We are here because we want to take a stand against Donald Trump. Millions of American women and men voted for him. Marching isn’t enough – we need to persuade, to win arguments, to challenge the causes of division and to build a future in common. For the sake of our children and grandchildren … we are here because we will not let the clock be turned back.”