In contravention of a court order, gardaí say, Aoife was taken from Swinford Health Centre in Co Mayo at approximately 10.10am on Tuesday morning.
In a statement, An Garda Síochána said it has serious concerns that there is an immediate and serious risk to the health or welfare of Aoife.
The Garda believes Aoife and her parents are in the Dublin city centre area.
Aoife has fair hair, and at the time she went missing was wearing a pink body suit with checked cream tights, pink Doc Marten boots and an animal-print jacket with a hood. According to the Garda statement, the jacket is printed with a number of animal prints including giraffes, zebras, monkeys and elephants.
Mr Murphy is described as being 5ft 6in, with a slight build and shaved hair.
Ms Haynes is described as being 5ft 8in, of slim build with long, straight, black hair.
The gardaí are asking that anyone with information relating to Aoife’s whereabouts contact them on 999 or 112.
The gardaí advise the public not to approach those involved, but to call 999, 112 or their local Garda station, providing as much information as possible.