Final Cut Pro video downloaded to flash drive
Without all the details of your hardware (both the Mac and flash drive), the most likely cause is the transfer speed available from the flash drive to your Mac. 

In this scenario you typically wouldn't opt to play the video from a flash drive, but rather just use the flash drive to transfer the video onto another device (such as your cousin's computer).

You should find that playing the video directly from a computer's internal drive, or from any modern external drive, should be fine. Note that by external drive I'm not referring to flash drives, but external HDDs or SSDs.

By way of a similar example, I recently had to produce an 8 minute video in FCP that needed to be viewable on a MS Surface device. It played fine on my Mac and fine on the Surface, but trying to play it from their 32GB flash drive resulted in a bad experience!