Is a sore, swollen black/purple bump on back of middle finger dangerous? That is, is it a blood clot that can travel? Or will it dissipate in time>

Depends. If this happens often, see your doctor. They may order some studies to look for a cause. If you recently bumped or injured it in some way it may be from that. However, it is not normal to have a bump that you describe. It would be best to have someone take a look at it today. Take a photo of it with your cell phone to help your doctor.
Possible. Bruise or hematoma from trauma are in the mix. Unlikely to migrate other than disperse a bit in the local area. Some history of trauma would suggest this. If no trauma, see a dermatologist to rule out a tumor mass.