Migrating from old iMac to new Mac mini with limited internal storage

You under purchased! You cannot use setup assistant or migration assistant. What you should do is return the 256gb

machine and get one with the same amount of storage as the iMac. Then use your Time Machine EXTernal Hard Disk to migrate your data, apps and settings.

I did not answer the latter part of your question. You could move your data files to the External Hard Disk but it would be a copy and paste production and leave you with a machine that under performs.

However as I already stated, you under purchased and in order to get the most out of your new MM you should return the 256GB machine, get one with the same or more storage as the old iMac. If you are short cash, then consider getting an Apple Refurbished MM which you can find in Apple's Refurbished store. You can find that by going to Apple.com then scroll to the bottom of the page and select the Refurbished store. There is ZERO downside to buying a refurbished Apple product, they are identical to new, have the same warranty as a new device, are eligible for AppleCare (highly recommended to buy !!!!) and save lots of money.

They aren't worthless but getting one is not a substitute for the MM's internal storage. Yes, you can purchase a 1TB refurbished MM for $1019 (specifically 24GB RAM and a 1 TB SSD), yes it's more than a 256GB model but it's the correct move. Follow the instructions I posted above to find it.

You are confusing value with cost, if you want low cost then stay with the solution you bought and suffer the consequences. If you want to be happy with the purchase you make then yes it will cost more money but the value is much higher.

FWIW I have always manually migrated after a clean install.

So I mount a Carbon Copy Cloner backup and use the Finder to copy my user files to the new install. I must also take care to copy some items such as ~/Library/Mail (and import that to the new Mail.app) and some custom preference and data files from ~/Libary/... (GraphicConverter, BBEdit, coconutBattery, WiFi Scanner) and VMware Fusion VMs from /Users/Shared so I don't have to rebuild them or lose that data. Then I install only really needed 3rd party applications.

Yes, that needs a tidy file system and knowledge where the important files are. Rest of the family uses CCC and Migration Assistant.

Your 256GB mini will be fine. I have 2 . . . an M2 and M4.

Do NOT use Migration Assistant as it is more likely to cause problems migrating rubbish!

Simply plug your new SSD/HDD into the iMac and manually copy everything you need.

You don't need to do everything at once . . . just copy the essentials and then as and when you need.

It is a very quick and simple process.

Considering your old iMac is running Catalina. it is likely that many of the apps will not be compatible with the new Mac Mini, especially 32 bit apps which are no longer supported for quite some time. Also, be aware that macOS will soon not support Intel apps on the Apple Silicon apps with Rosetta2 as that will be soon to go away in the next coupe years.

So, putting your data files on an external drive and manually re-installing apps on the new Mini (the ones that still may be compatible) is the best approach. You will likely need to manually set your preferences in the new install. This may not be as bad as it sounds but the fresh install can help to remove old baggage that has been carried along by the apps for years that is really no longer of any use. This also gives the option of getting rid of all the junk that has built up from the old iMac.

Things like Music, Movies, and Photos libraries can easily be put on the external drive and all you need to do is point the apps to their new location.

Another thing, you will also need a second hard drive to back up all your data, unless your data is of little importance should the hard drive fail.