How do I update to a specific version? Stability over novelty required

You do not have to upgrade your Mac Pro to 11.4 to use Migration Assistant.

I am unclear what you are attempting to do here.

What macOS version is currently running on the iMac?

What version do you want it to be running?

If it isn't already, your iMac can easily run macOS 11 Big Sur. The upgrade path to the current version – 11.6 – is easy and described below.

Do NOT upgrade until you have made a Time Machine or other backup of your startup drive.

How to Upgrade  macOS:

The macOS version you can upgrade to depends on the year your Mac was released.

Click the  (Apple) menu at the top left of your screen and choose About This Mac.

macOS 11 Big Sur is compatible with the Macs listed here:

https://support.apple.com/en-us/HT211238

❗️ – IMPORTANT – Backup your data, now, before you attempt an upgrade.

Learn How to Back up your Mac with Time Machine:

https://support.apple.com/en-us/HT201250

❗️ – IMPORTANT – NONE of your 32-bit apps will work in Catalina or Big Sur.

https://support.apple.com/en-us/HT208436

Before you upgrade, use this free utility to check the compatibility of your apps:

Go64: https://www.stclairsoft.com/Go64/

🖥 – Check the compatibility of your Mac with newer versions of macOS.

https://support.apple.com/en-us/HT201475

✅ - Using Safari only, download the macOS upgrade installer app:

Click > Big Sur.

Once downloaded, double-click the installer app in the Applications folder to begin the upgrade.

No, Apple cannot 'send' 11.4 to your iMac. Apple only makes the newest point release of each OS available for download.

The best solution for stability is to run the previous major version of MacOS... currently Catalina 10.15.6. If that works you can find download instructions here: https://support.apple.com/en-us/HT211683

As a creative work around to 11.4... since you currently have a computer with 11.4 you -might- be able to install 11.4 from the recovery partition from your 2013 MacPro if you first downgrade the 2020 iMac's Secure Boot to Medium and allow external boot. https://support.apple.com/en-us/HT208198

As a 3rd alternative, just try running your audo software on 11.6 and seeing what happens. Maybe you will reproduce whatever known bugs exist, or maybe it will just work.

P.S. If you actually meant 10.14 is required for software with 32-bit components then that is an insurmountable problem. 2020 iMac hardware is too new to have 10.14 drivers.

What system did it have when you purchased it?

Yu can't install an earlier system on the 2020 iMac than it came with. It's just not permitted.

Therefore you'll need to either continue using your MacPro until the apps are updated for current macOS systems.

If you bought the 2020 iMac from Apple you have 14 days from the date of purchase to return it no questions asked for a full refund.

You have good advice above. I will offer some thoughts on stability. It is my experience that most operating systems are quite stable. That is why they are first beta tested extensively. Instability is generally due to things that are under a user's control. Either they fail to do important maintenance which includes such things as emptying the Trash, periodically restarting the computer, occasionally restarting in the safe mode to "fix" things and occasionally running Disk Utility's Repair function. More important, some people install junk software that turns out to be incompatible with other software. One example of this are antivirus apps. In summary, the best defense is to learn about our Macs and then do those things which can keep it healthy and running well. Remember, software is very complex. I say keep the computer clean and only install things you actually need.

IMHO Big Sur is plagued by heavy problems on third party drivers. Latest Catalina 10.15 is the best choice. Be careful with what already advised above especially regarding having 1 working backup at least (if you upgrade from Mojave better have a 2nd fresh backup, for more security in case of needed downgrade) and 32 bit apps/drivers/plugins not working in Catalina. Have doubt on this? Download Go64 and check your sw and therefore, if needed, possibility to update to 64 bit version.

(Answer to Brendenden)