No, MA only goes into the MM itself, you cannot migrate to an External Hard Disk. If your old Mac has 1 TB of data on it (this is just an example) and the new Mac mini only has 256GB of internal storage Migration Assistant will NOT work. You simply cannot put 1 gallon of milk into a 1 cup container!! That's just common sense, you can try to put 1 gallon of milk into a 1 cup container however what you will end up with is a huge mess. The same holds true with migrating data!
The new MM should have at least the same amount of storage as the old Mac. It is unwise to try anything else, yes we know it's more expensive but that's the ONLY way its going to work!
If my hunch is correct (that you have more data on old Mac than the new MM has room for) then you will have to setup the MM manually and try to copy data files from the old Mac to the External drive. That will be a very slow and messy process. If my hunch is correct, please return the MM and buy one with the correct amount storage.