The campaigners want media outlets, including the Mail, the Express and the Sun, to end their negative editorial stance towards refugees.
It has created an organic social media campaign urging brands to pull their advertising revenue from newspapers that report negatively on foreigners.
In a major victory for the campaign, Lego announced on Twitter over the weekend that it would end a regular promotion with the Mail, in which a voucher can be exchanged for toys, with no plans to continue it in the future.
Lego said it had listened very carefully to its customers after a well-reported open letter from a father to Lego, which was published on Facebook, accused it of betraying its own brand vales with the partnership.
Bob Jones wrote: "For a few years now you have done free giveaways in the Daily Mail newspaper.
And while holding back that wretching feeling, I've paid for a copy to get the free Lego pack... but lately their headlines have gone beyond offering a right-wing opinion. Headlines that do nothing but create distrust of foreigners, blame immigrants for everything, and as of yesterday are now having a go at top judges in the UK for being gay while making a legal judgment. It genuinely bothers me, that a great progressive company like yours supports this "news" paper, helping increase its circulation."
John Lewis responded to the campaign on Friday. A spokesman said: "We fully appreciate the strength of feeling on this issue but we never make an editorial judgement on a particular newspaper."
A Walkers spokesman said: "We have a very successful partnership with Gary Lineker and we will continue to do so. Our advertising approach is not determined by the editorial stances of individual newspapers."
Lego’s decision to end the partnership with the Mail will surely leave the media outlet rattled, the Guardian’s media commentator Roy Greenslade told PRWeek.
He said: "Although it’s only one case, it must be worrying. Remember that what persuaded Rupert Murdoch to close the News of the World in 2011 was not phone-hacking itself but the fact that advertisers responded by pulling their ads. With newspaper finances already precarious, if companies follow Lego then there could be trouble ahead."
According to Brandwatch, there were nearly 60,000 mentions of the #StopFundingHate hashtag on Twitter
between Thursday and Monday morning at 10am, with mentions split almost evenily between men and women. There were also more than 26,000 mentions in conjunction with John Lewis' #BusterTheBoxer hashtag
, 11 per cent of the total mentions for John Lewis.
There were also 76,000 mentions involving the Daily Mail and LEGO in the same tweet.
Will McInnes, CMO of Brandwatch, said: "Given just how many brands and voices are competing for attention in the Christmas ads season, for the #StopFundingHate campaign to successfully hijack 11% of mentions of the John Lewis campaign is significant. With such apparent momentum it's no surprise that a politically conscious company like Lego felt compelled to take action."
Campaigners now feel emboldened to carry on targeting brands that advertise in the Mail, Express and Sun, following the weekend’s success.
Stop Funding Hate's Wilson said: "We fully support freedom of expression - and not only for newspapers but also for customers like Bob Jones, whose polite, friendly and heartfelt message Lego has now taken on board. Bob Jones deserves huge credit for this result - as do the thousands of people who shared his Facebook post and who have supported the Stop Funding Hate campaign on Facebook and Twitter.
"Lego have made a brave and principled move by responding to customer concerns on this issue. Now thousands of people are calling on the Co-op, John Lewis, and Waitrose, and Marks and Spencer to do the same. We very much hope that they will listen."
The Daily Mail was asked to comment but had not responded at the time of publication.