Here's just two of their distressing stories.
in the Scottish Refugee Council's centre in Glasgow, there are more than a dozen people in the waiting area. But with the asylum advice services now officially winding down, it's a relatively quiet day - usually the queue stretches out the door.
Among those seeking help is an Iranian in his late twenties who has recently arrived in Glasgow from Leeds. He has been refused asylum, is destitute - and has cancer. He can apply for emergency support - comprising accommodation and around £35 a week - on medical grounds as he is unfit to travel and leave the country. But he cannot register with a GP without an address and has no idea where to find a doctor to fill out the required form.
A case worker makes him an appointment with the homeless health service in Glasgow. As he leaves, he is also handed a note to give to a local charity which will be able to give him clothes and shoes.
A pregnant woman in her twenties from Iraq arrives with a letter from the Home Office to ask what it is about. The case worker explains the letter is for an emergency support token which "bridges the gap" before long-term support is set up. She also is given help filling in a form for a maternity grant - a one-off payment of up to £300 when the baby is born and £3 extra a week eight weeks before the due date and six weeks after.
The countries where most asylum seekers currently come from are Iran, Eritrea, China and Syria. Esther Muchena, the asylum services manager at the Scottish Refugee Council, says there are a wide variety of issues to deal with, ranging from help with accommodation to assisting new mothers and victims of crime.
Falling into destitution can become a problem at any point during the asylum process. "An administration error by the Home Office can mean someone is left without support," Muchena says.
The Scottish Refugee Council will continue to provide specialised advice to families with children up to eight years old, and will help unaccompanied young people going through the asylum system. But most others will need to turn to the new system, run in Kent.
"It is going to be much more difficult for people if they have to phone for advice," one worker says.
"Sometimes it is about offering a glass of water, making sure people are warm and comfortable - these things that you do as a human being that will be lost."