It sparked a major police operation in the town last week before the man was eventually traced in Scunthorpe.
The public was warned not to approach the missing Midpark hospital patient and children kept in some schools.
NHS Dumfries and Galloway said it would be looking at the circumstances of the incident and aim to "minimise the likelihood" of it being repeated.
A statement from the health board confirmed it would be looking into how the patient had been able to abscond.
"NHS Dumfries and Galloway have well established processes for undertaking reviews of incidents that occur," it said.
"As per these guidelines, a review of the event will be undertaken."
However, it said that there had been no immediate changes to operations at the unit.
"NHS Dumfries and Galloway have well established standard operating procedures in place that are regularly reviewed in line with national legislation," said the statement.
"However, no known changes have been made to the procedures following the event."
The health board added that it would make every effort to avoid any similar incidents.
"NHS Dumfries and Galloway can give assurance that standard operating procedures are in place and regularly reviewed to comply with national legislation around the treatment offered to patients, while maintaining a person-centred approach," concluded the statement.
"The procedures will aim to minimise the likelihood of such an incident re-occurring."
A health board is to carry out a review after a patient absconded from a Dumfries mental health unit.