How do I record and play a series of videos in some software-controlled order?

I’d like to create an interactive artwork where people can come up to a camera and record a short video of themselves speaking. At the same time, there will be a series of such videos playing on a nearby display. I should be able to insert the new video at any point in the list of playing videos. The list should also be continuously changeable via software; any programming language will do. Bonus points if I can have basic transitions between the videos.