Any animals nearby will walk with you, yes. You can also use this to move animals between edges of maps.
Thank you so much for the comments, it's really appreciated. :D Yes, I am still working on this. I have an update coming soon (1.2.1) with some small additions and bug fixes.
EDIT: I was testing around and figured the Foodmix actually had a bug where animals would not follow properly. This will be fixed in the next patch, which should be available in a short time.