I found the esri/geometry/screenUtils and used the functions to implement this. Not sure if it's the most efficient way but it works.
// Moving a drawing also moves the associated measures
this.editToolbar.on('graphic-move-stop', lang.hitch(this, function(event){
if (event.graphic.annotation && event.transform){
var measurementScreenPoint = screenUtils.toScreenGeometry(this.map.extent, this.map.width, this.map.height, event.graphic.annotation.geometry);
measurementScreenPoint.setX(measurementScreenPoint.x + event.transform.dx);
measurementScreenPoint.setY(measurementScreenPoint.y + event.transform.dy);
event.graphic.annotation.geometry = screenUtils.toMapGeometry(this.map.extent, this.map.width, this.map.height, measurementScreenPoint);
this.drawGraphicsLayer.redraw();
}
}));
Note that the event.graphic.annotation is a Graphic reference I added myself and represents the measurement Graphic I created.
Better ideas are welcome.