Intrafraction tumour (e.g. lung) motion due to breathing can, in principle, be compensated for by applying identical breathing motions to the leaves of a multileaf collimator (MLC) as intensity-modulated radiation therapy is delivered by the dynamic MLC (DMLC) technique. A difficulty arising, however, is that irradiated voxels, which are in line with a bixel at one breathing phase (at which the treatment plan has been made), may move such that they cease to be in line with that breathing bixel at another phase. This is the phenomenon of differential voxel motion and existing tracking solutions have ignored this very real problem. There is absolutely no tracking solution to the problem of compensating for differential voxel motion. However, there is a strategy that can be applied in which the leaf breathing is determined to minimize the geometrical mismatch in a least-squares sense in irradiating differentially-moving voxels. A 1D formulation in very restricted circumstances is already in the literature and has been applied to some model breathing situations which can be studied analytically. These are, however, highly artificial. This paper presents the general 2D formulation of the problem including allowing different importance factors to be applied to planning target volume and organ at risk (or most generally) each voxel. The strategy also extends the literature strategy to the situation where the number of voxels connecting to a bixel is a variable. Additionally the phenomenon of 'cross-leaf-track/channel' voxel motion is formally addressed. The general equations are presented and analytic results are given for some 1D, artificially contrived, motions based on the Lujan equations of breathing motion. Further to this, 3D clinical voxel motion data have been extracted from 4D CT measurements to both assess the magnitude of the problem of 2D motion perpendicular to the beam-delivery axis in clinical practice and also to find the 2D optimum breathing-leaf strategy. Issues relating to the practical calculation of the strategy, including effects on leaf velocity and effects of different spatial-sampling frequencies, have been investigated, and unattenuated-fluence maps have been produced showing the effects of the differential motion and tracking. It was discovered that large distances between adjacent leaf-ends could cause the tracking to fail when there was tissue motion across the leaf channels. To overcome this problem the use of 'synchronized' leaf trajectories, which ensure that adjacent leaf-ends are always close enough to each other to facilitate tracking, has also been investigated.