Where does a large file go when I delete it from a volume?

List the volume's hidden folders in the Terminal with:

$ ls -laht /Volumes/
...
d-wx-wx-wt    3 mmorin  staff   102B 12 Sep 09:35 .Trashes
...

The directory .Trashes is what you're looking for, which is why .Trash does not exist.

In my case, I needed sudo to view the directory, which explains why /Volumes/<volume name>/.Tr did not complete. Nested under the directory 501 are the deleted files:

$ sudo ls -lahtr ../.Trashes/501
total 15943672
-rwxrwxrwx@  1 _unknown  _unknown   1.0G  3 Jan  2018 some file.MP4
-rwxrwxrwx@  1 _unknown  _unknown   999M  3 Jan  2018 another file.MP4
...