I want to move Final Cut Pro and Logic Pro X from Intel Mac mini 2018 to a Mac mini M4?

For Logic Pro you'll need to re-download the app from the App Store to install it on your new Mac mini.

Then download any components of the sound library fresh from within the Logic app on that Mac.

As for any Logic projects that you have saved on the old mini, those can be migrated to the new Mac with the rest of your data from the old one. Or you can transfer them manually using an external drive or Time Machine backup.

For FCP:

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To move your Final Cut Pro (FCP) library and projects to a new Mac, you'll need to copy the library file to an external drive, then move that drive to the new Mac and open the library from there. 

Here's a more detailed breakdown:

1. Prepare Your Library:

• Consolidate Media:
• Ensure all media files linked to your projects are within the library itself. This can be done by going to the Library Properties inspector, selecting "Media," and choosing "In Library". 
• Delete Generated Files:
• Optionally, to reduce the size of the library, you can delete generated files like render files, optimized media, and proxy media. 
• Deactivate Third-Party Plugins:
• If you have any third-party plugins, deactivate them on your old Mac before moving the library. 

2. Copy the Library:

• Locate the Library File: In Final Cut Pro, select the library in the Libraries sidebar, then choose File > Reveal in Finder. 
• Quit Final Cut Pro: Ensure Final Cut Pro is closed before copying the library file. 
• Copy to External Drive: Drag the library file to your external hard drive. 

3. Transfer to the New Mac:

• Connect External Drive: Connect the external drive to your new Mac. 
• Copy to New Mac: Copy the library file to the desired location on the new Mac. 
• Open in Final Cut Pro: Double-click the library file to open it in Final Cut Pro on the new Mac. 

4. Reinstall Plugins:

• Install Third-Party Plugins: Reinstall any third-party plugins on the new Mac using the installers provided by the manufacturer. 

5. Final Cut Pro License:

• App Store:
• If you purchased Final Cut Pro from the App Store, you can download and install it on your new Mac using the same Apple ID. 
• License Agreement:
• The Final Cut Pro license agreement allows you to use the software on Apple-branded computers that you own or control. 

You're welcome.

You can simply copy FCP and Logic to your new Mac to save time downloading.