The charity blamed its financial crisis on late payment of legal aid by the Legal Services Commission. It had asked the government to alter the legal aid system to speed up payments, but this request was refused.
The demonstration is taking place outside the Ministry of Justice in London in a bid to change ministers' minds and avert the permanent closure of the charity next week.
Rachael Maskell, Unite national officer, said: "With the emergency Budget just days away, George Osborne could save this organisation. Government has a simple choice, they could walk away or show that they really value charities."
Unite says 300 solicitors, legal advisers and other staff, as well as the families the charity supports, will be affected by the closure of the charity.
"Many of the RMJ clients will face being returned to countries where they face persecution and where their lives will be put in danger," Unite added in a statement.
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