Hi Larry,
Without knowing the detail of your local datum, I tried using this case: UTM (WGS84 datum) as the projection of the mosaic dataset and NAD27 and NAD83 as datum for the images that will be loaded to the mosaic dataset.
1. create a mosaic dataset with UTM projection and add datum transformations from the GP environment setting for transformation between WGS84 and NAD27 and WGS84 and NAD83, there are many options that are available, but make sure that you pick one that is suitable for your data
2. add the images to the mosaic dataset. This should be done. when accessing the mosaic dataset (the images in the mosaic dataset), the system will pick the corresponding transformation to use.
when you comparing with ArccMap, please make sure that they are using the same transformation (ArcMap default may not be the one that you pick from the environment setting). If you still see issue or issue with your data, please contact Esri support with your data to reproduce.
Regards,
Hong Xu