Windows 10, GTX780
Broken: 2.79 5bd8ac9 with --enable-new-depsgraph
Worked: 2.79 5bd8ac9 with old depsgraph
Short description of error
Using Blender 2.79 with new Depsgraph activated on launch makes the Relations Extras option "Slow Parent" stop working. The "Offset" parameter to determine how many frames of delay the parent relationship will have simply have no effect at all, and the Child object will follow without any delay at all.
Exact steps for others to reproduce the error
Open 2.79 using a bat file with steps to enable New Depsgraph (look online for instructions). Create two objects, parent one to another, and set the Child object to have a Slow Parent under the Relations Extras section of the Object Data tab. Set an Offset value that is greater than 1 and move the parent object around. You should see the Child object move delayed in relation to it's parent (do the same steps on 2.79 default launch), but this doesn't work with New Depsgraph activated.
I'm aware that blender 2.8 will work based on this/a new Depsgraph also... therefore I think it's important to make sure that the reason the Slow Parent stops working under the new Depsgraph is known, so that on 2.8 this problem doesn't persist.
This Slow parenting option is very useful for certain cases where one would need to do transformations to the Child position in it's Local Space using a Constraint for example. If you use Drivers instead to emulate Slow Parent, the constraints are kind of ignored because Drivers have priority over constraints I think.
Edit: Sample (working) file uploaded. Use the "--enable-new-depsgraph" command on a new blender instance to see that the slow parenting option breaks.