[Drag select keeps selecting a single item]

Trying to move multiple items after drag select was driving me crazy until i realised what was actually happening.

Using the move tool I drag over several objects which are then shown as selected with a wireframe. NOTE that one of the selected items has handles.
Click and drag the item with the handles to move the selection around the canvas. If you don't click the and drag the the item with handles it will de-select all and only select the one you clicked on to drag.

Grouping is another option to drag multiple objects, but your need to group items will vary.

It might be a good idea to make sure the preference/tools - "Select object when intersects with selection marquee" is ticked if you want a more relaxed selecting objects experience.