Can migrate assistant transfer individual files?

Nope, MA is an all or nothing proposition. You can tell it to migrate:

• Settings
• Apps
• Data

However that is about it. If you want a clean migration then clean up the old Mac, backup using Time Machine and follow the steps in Setup Assistant and Migration Assistant for a dead reliable and fastest possible migration.

My Migration Assistant on my new MacBook Pro freezes at various stages during my attempt to migrate from my old iMac using the Time Machine backup on a 4 Tb USB external HD (last backup last evening). One time it transferred most data over several hours but then froze during setup with '13 minutes to go'. Today it freezes when I attempt to even start a new MA process to complete the transfer. My new MacBook is useless until I can complete this. Help!

Why not use iCloud or an external drive and do copy/paste?