When introducing "drag-all-selected" support all over Blender, we figured this wouldn't work well with the Graph Editor selection/transform behavior.
Hence, William and I have been working on the following changes, although we used this chance to improve the behavior in general too.
For more info see T70634.
* Handles now always move with the key, regardless if they are selected or not.
* Selecting the key doesn't select the handles anymore, their selection is separate.
* Multiple keys and handles can now be dragged.
* Dragging a handle moves all selected handles on the same side.
* Tweak-dragging any handle can never affect any keyframe location, only handles
* G/R/S should behave as before.
* Changing the handle type with a key selected always applies the change to both handles.
* Box selection with Ctrl+Drag now allows deselecting handles.
* Box selection Include Handles option now only acts on visible handles, wasn't the case with Only Selected Keyframes Handles enabled.
* Box selection Include Handles is now enabled by default in all bundled keymaps.
If needed I can give more detailed rationales for each change, but keeping this short.
The changes have been tested for some days by the animators here in the Blender Animation Studio. Some changes are based on their feedback.