Labour leadership candidates who signed a pledge calling several organisations “trans-exclusionist hate groups” are facing demands to produce evidence for the allegation.

A row over a pledge card drawn up by the Labour Campaign for Trans Rights group broke out last week after Rebecca Long-Bailey, Emily Thornberry and Lisa Nandy, as well as deputy leadership candidates Angela Rayner and Dawn Butler, all expressed support for the charter. It calls on Labour to expel “transphobic” members, and describes campaigns including Woman’s Place UK as “trans-exclusionist hate groups”.

Woman’s Place UK has now written to the leadership figures demanding to see the evidence behind the claim. The group has also written to the candidates who have not signed the pledge, asking to meet them to discuss the growing row. It said none of the candidates had yet replied.

The Labour Campaign for Trans Rights pledge states that the party has “failed to act as transphobia has gained ground” within the party. It said last week it was “heartened by the solidarity” of the Labour candidates. It reiterated its view that Woman’s Place UK perpetuated discrimination against trans people. It is the latest development in a growing row within the party. Woman’s Place UK is pushing for ministers to consult more widely about changes to the Gender Recognition Act, which would allow people to legally self-identify as a man or a woman without medical approval.

It rejects accusations that it is transphobic and trans-exclusionist. Its founders say it aims to “ensure that women’s voices are heard and our sex-based rights upheld”. However, critics say it is trying to limit the rights of trans people.

In a letter to Lisa Nandy, Ruth Serwotka, a Woman’s Place UK founder, wrote: “I hope you understand that expulsions of women from the party on the grounds you are suggesting (and being cavalier about evidence) will provoke a huge backlash. It will be fiercely opposed by all fair-minded members, who insist on reason and evidence.

“In fact, we are a women’s rights organisation committed, among other things, to upholding current sex discrimination and equality law as set out in the Equality Act (2010), including upholding protections for trans people”.

Nandy’s team declined to comment. The government has delayed publication of its long-awaited decision on changes to the Gender Recognition Act.