The row over debates surrounding transgender and women’s issues arose at Lancaster City Council last week, ahead of Sunday's Pride event in the city.
However, Coun Dowding denied any wrongdoing, defended her right to question and to disagree in debates, and defended her record on diversity and support for Pride.
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Coun Oliver raised his allegations at the full council meeting, during a report which included references to increased anti-social behaviour in Lancaster, including evidence of victimisation and hate crime.
Earlier this year, Coun Robinson, a university ward representative, raised a number of reports about incidents of abuse and assaults, including assaults on foreign and Jewish students in Lancaster.
Coun Robinson said: “A few weeks ago, I was surprised to see a rather shocking article re-tweeted by Coun Dowding. Many councillors can be caught out by sharing articles they haven’t read closely.
"But I think its clear that Coun Dowding shared this deliberately and it seemed to be a rather openly problematic article from the outset. She was then challenged and became involved in debate on Twitter.”
Recent Daily Mail content has included debates about gender identity, views on whether there are only male or female bodies, trans activism, and the work of different groups including the charity Stonewall.
Some content has referenced a legal dispute in which a person called Allison Bailey claims she has lost work because she had criticised Stonewall’s trans policies.
In one opinion feature, broadcaster and former Stonewall figure Simon Fanshawe claimed Stonewall’s work over the years is being ‘squandered’ by current trans activists — but not all trans people, he emphasised.
Fanshawe has also raised questions about debates on ‘male bodies’ in women’s safe spaces, women’s changing rooms and sport, and he claimed celebrities such such as J K Rowling or Martina Navratilova were told to ‘shut up’ if they questioned current thinking.
Coun Robinson claimed Allison Bailey is a co-founder of the LGB Alliance, a group allegedly criticised for its views by others such as the LGBT+ Liberal Democrats, the Labour Campaign for Trans Rights and Carla Denyer, the Green Party co-leader.
He wanted to know Coun Dowding’s views on the LGB Alliance and Stonewall - but he said he would forward his full question and screenshots to the leader, which he also supplied to the Local Democracy Reporting Service.
In reply at the meeting, council leader Caroline Jackson said: “I would not like to suggest that we do not support trans rights, but I think there are quite a few people who get rather lost about this, so I will give you a written answer.”
Coun Dowding later told the Local Democracy Reporting Service she felt some of Simon Fanshawe’s recent comments were useful about breaking down prejudice and building alliances with critics.
Fanshawe had argued, she felt, this was achieved in the past ‘not by screaming abuse or attempting to lay down the law but by using data and research’.
He advocated ‘celebrating our differences, not wiping out diversity’, she added.
Regarding her Twitter activity, Coun Dowding said the screenshots showing debate between her and others circulated by Coun Robinson related to a different article.
She added: “Coun Robinson has alleged I shared a transphobic Daily Mail article. The article he highlighted in my re-tweet refers to an opinion piece by Simon Fanshawe, a co-founder of Stonewall.
"I think Simon Fanshawe makes some interesting points. People can decide for themselves.
“I was surprised that Coun Robinson thought it was constitutional, or otherwise appropriate or reasonable, to use the time for questions about the leader’s report to try to hold the leader to account for my re-tweets.
“However, in a week in which trans people have been facing a barrage of abuse, I am saddened that Coun Robinson has tried to politicise this issue, and that he didn’t approach me about it directly.
“I am an ally of the LGBTIQA+ community and I am deeply proud of Lancaster Pride, and the city’s history as a welcoming and open city for all.”