Migration Assistant between Mojave and Catalina

No, there's no need to upgrade the older Mac if you are migrating from old to new.

Hello

I understand you want to know if the software on both Macs needs to be updated in order to use Migration Assistant. I am happy to provide you with information about Migration Assistant.

"Check software, settings, and power

• Install all available Apple software updates on both Mac computers. Install any updates for your third-party apps as well.
• Make sure that your old Mac is using OS X Lion or later. 
• Make sure that your old Mac has a computer name: Choose Apple menu  > System Preferences, then click Sharing and check the Computer Name field.
• Connect both computers to AC power."

How to move your content to a new Mac

Thanks for using the Apple Support Communities. 

Take care.