[rekordbox for Mac/Windows]

Yes.

Use the recording function of the DJ application connected to the DDJ-FLX4.

Click the [REC] button at the top left of the rekordbox screen to display the recording panel in the middle of the screen.

Click the [REC] button in the panel to start/stop recording.

For details, see [Recording] in [Performance mode] in the rekordbox Instruction Manual.

See the rekordbox, Instruction Manual.

[For Serato DJ Lite]

No. This isn't possible when using Serato DJ Lite.

You can record performances when using Serato DJ Pro, but you'll need to buy a Serato DJ Pro license or take out a subscription for the software.

[For Serato DJ Pro]

Click the [REC] icon at the top left of the Serato DJ Pro screen to display the recording panel in the middle of the screen.

Click the [REC] button in the panel to start/stop recording.

For details, see the Serato DJ Pro software manual at serato.com.