The exhumation happened on Tuesday and it said the formal process to establish the identity of all of the remains found in the grave has begun.
“This process may take some time,” it added.
A former Cistercian monk from the Beechmount area of west Belfast, Mr Lynskey later joined the IRA.
Mr Lynskey went missing in 1972, and republicans have claimed he was “executed and buried” by the IRA.
The latest search for his remains was in 2018. It ended without success.
Last November his niece said that his body “needs to be brought home”.
So far, the remains of 13 of the Disappeared have been recovered.
The remaining four are Mr Lynskey, Columba McVeigh, Seamus Maguire and Robert Nairac.
The disappearance of Mr Maguire was taken on as a new case by the Independent Commission for the Location of Victims’ Remains (ICLVR) in 2022.
The ICLVR was established by the UK and Irish governments in 1999.
Annyalla is between Monaghan town and Castleblayney.
