It is conjectured that the full (spacetime) Bartnik mass of a surface Σ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by Σ. Assuming this holds true for a 1-parameter family of surfaces Σt evolving in an initial data set with the dominant energy condition, we compute an expression for the derivative of the Bartnik mass along these surfaces. An immediate consequence of this formula is that the Bartnik mass of Σt is monotone non-decreasing whenever Σt flows outward.