However, now that 10.4.0 is with the release team (AKA no more development changes), there are some vertical transformations possible in ArcGIS for Desktop 10.4.0. It's currently for vector data only.
The Project tool has a check box to enable vertical transformations. The input data must have vertical data and a vertical coordinate system defined. When you set the output coordinate systems, you can now set both the horizontal (geographic/projected) and vertical coordinate systems.
We look up a combination horizontal and vertical transformation using a table, hvtdefaults.json, which is in the pedata folder of the install. It's known to be incomplete. If it comes up with transformations that you don't agree with, you can either edit the json file or try to perform the projection/transformation in multiple steps.
Vertical transformation data tends to be in grid files which are large. For 10.4.0, there's now a separate ArcGIS Coordinate Systems Data setup which can optionally install the US geoid2012 files and VERTCON files (NGVD29 / NAVD88), and the world EGM84 and EGM2008 1'x1' or 2.5'x2.5' geoid models.
There's some supporting functionality in ArcObjects too.
Here's a picture:
Melita