How do sunspots move seen from the side? Sunspots rise vertically? Do they float on the surface like oil? Do they flow up or down vertically from the Sun's core like a lava lamp?
The sunspots are actually depressions and low thermal pressure zones. 

Incredible magnetic fields are piercing the surface of the sun and the magnetic force pacifies the movement and the convection at the zone of the sunspot. that's why the surface becomes darker clouds near those magnetic fields. 

Elsewhere, the sun's surface has convection wells called solar granules a bit like a lava lamp, which are about 1000km diameter. You can check images of "solar convection" to study maps of convection. 



At a sunspot the movement of colder plasma flows downwards, and there is a hot spot trapped under the sunpot where the thermal energy escapes radially away from the sunspot. 



This is a video attempting to illustrate the view under the sun:

https://www.youtube.com/watch?v=zU9O8SmCHFQ

This is a virtual simulation of them. The versions are wildly different: 

https://www.youtube.com/watch?v=W5dldC5dPMk

https://www.youtube.com/watch?v=UN_9HMVXjY0

https://www.youtube.com/watch?v=ARIJx0geaOE

here is a speeded up video of some sunspots:

https://youtu.be/BejRxZAa66E

https://youtu.be/l1VpMRMg4P4

https://youtu.be/T5MnZUINzjY