A Jamaican man deported from the US to Eswatini under President Donald Trump's tough new immigration rules has been sent on to Jamaica, the Eswatini government has said.

Orville Isaac Etoria, 62, was voluntarily repatriated at the weekend, and was "warmly welcomed by members of his family", it added.

But his treatment has been condemned by the Legal Aid Society of New York, which says Etoria came to the US as a child and had lawful permanent resident status there for decades.

Etoria and four other individuals - from Cuba, Laos, Vietnam and Yemen - were described by the US government as "depraved monsters" when they were expelled to Eswatini in southern Africa in July.

Since then, they have been held in solitary confinement in a maximum security prison in the capital city Mbabane.

Lawyers for the remaining four deportees say they have not been able to contact them. But Eswatini's government has said that it is engaging in efforts to repatriate them all.

Trump's pledge to conduct mass deportations was a centrepiece of his election campaign and an issue on which he drew widespread support.

In some cases, people are expelled to countries where they have no ties.

Etoria had "transformed his life" while serving a 25-year prison sentence for murder and reintegrated into American society, when he was deported and imprisoned again in Eswatini without due process, the Legal Aid Society society.

Activists in Eswatini also denounced the deal as "unconstitutional" and protested outside the American embassy.

Rights groups have instituted legal action in a bid to overturn it, saying the government could not strike the deal out without the approval of parliament.

The government said it had acted within its powers.

Eswatini is the last absolute monarchy in Africa. Previously known as Swaziland, the a small, landlocked country is surrounded by South Africa and Mozambique. It has been led by King Mswati III since 1986.