On the eve of the first same-sex marriages in the UK, the Australian Greens have announced that they will introduce a bill giving legal recognition to international same-sex marriages in Australia.

"Gay and lesbian Australians shouldn't have to check their marriage licenses at the customs gate," the Greens' LGBTI spokesperson, Senator Sarah Hanson-Young said.

"While some of our closest allies are busily reaping the tourism rewards that come with marriage equality, Australia is stuck in the past.

The Australian Greens want Prime Minister Tony Abbott to join President Barack Obama and others around the world in condemning new laws in Uganda that outlaw homosexuality with sentences of life imprisonment.

"Tony Abbott has a moral obligation to condemn these horrific laws and to work with the global community to guarantee the safety of all those named as LGBTI in the Ugandan tabloid Red Pepper," Greens Leader Christine Milne said.

Greens Deputy Leader Adam Bandt says the Greens are ready to co-sponsor a new marriage equality bill but has warned Labor against jumping the gun before a cross-party committee does its work.

Mr Bandt has written to Labor's Deputy Leader Tanya Plibersek asking her to allow the cross-party committee to reach a consensus on the way forward before proceeding with a bill.

"The best way of getting a bill through Parliament is to let the cross-party committee to do its work and achieve a consensus bill that can be sponsored across the Parliament," Mr Bandt said.

A Greens' motion supporting Australian athletes at the Sochi Winter Olympics who have spoken out in support of the LGBTI community has passed the Senate today.

"Today the Senate showed its support for Australian athletes who are standing for the LGBTI community in Sochi," the Greens' LGBTI spokesperson, Senator Sarah Hanson-Young said.

"The Olympics are about celebrating human achievement. They're about inclusivity and excellence, they're not about discrimination.

The Australian Greens will move a motion in the Senate congratulating all Australian athletes who are standing up for the LGBTI community while attending the Sochi Winter Olympics when Parliament resumes.

"The Olympics are about bringing people together, not discrimination," the Greens' LGBTI spokesperson, Senator Sarah Hanson-Young said.

"It's important for governments and parliaments around the world to congratulate brave athletes who stand up for what's right, and that's exactly what we'll be doing.

The Australian Greens say that the High Court decision today on the ACT's marriage equality laws is incredibly disappointing, but that ending marriage discrimination is inevitable.

"Our hearts go out to the couples married in the last week who will be devastated by today's decision. My message to those couples is this: we will never ever give up until marriage discrimination is a relic of history," Greens Leader Christine Milne said.

A Federal Cross-Party working group on marriage equality has been announced today featuring the Greens Senator Sarah Hanson-Young, Labor Senator Louise Pratt and Liberal Senator Sue Boyce.

"Federal marriage equality got a lot closer today," the Greens' LGBTI spokesperson, Senator Sarah Hanson-Young said.

"It's only by working together that we're going to achieve this essential reform.

"Equality minded members of the Labor, Liberal and the Greens parties will now have to do everything they possibly can to get marriage equality across the line.

The Australian Greens have given notice today that they will introduce a marriage equality bill into the Senate and have invited members of all parties to co-sponsors the legislation.

“Today the fight for federal marriage equality has started a new,” the Greens’ LGBTI spokesperson, Senator Sarah Hanson-Young said.

“Marriage equality is no longer about any one person or any one party and that’s why I’m asking members of all parties to co-sponsor this bill.

Australian Greens Leader Christine Milne says the Abbott government's wasteful move to strike down ACT marriage laws in the High Court will be nothing more than a futile attempt to maintain discrimination against same-sex couples contrary to the wishes of the Australian people.

"The Abbott government should remove marriage discrimination itself and save the States and Territories the effort and cost," Senator Milne said.

"The Greens will be introducing legislation yet again when the Parliament resumes and I invite the PM and Labor to get behind it.

The Australian Greens say Attorney-General George Brandis should release legal advice he says justifies High Court action against ACT marriage equality laws.

"Attorney-General George Brandis should put up or shut up when it comes to the advice that supposedly justifies spending public money on a legal challenge that is all about maintaining last century discrimination," Australian Greens Leader Senator Christine Milne said.