You can achieve that using the custom roles from kanboard. But it will apply to the whole project, not just to specific tasks.
Project > Configure this project > Custom roles > Add new custom role > name it
Click on the new role > Add new drag and drop restriction
Select a source column, select a destination column, choose if you wish they can only move assigned tasks, save.
Despite there is a “source” and a “destination” column, the reverse path is also allowed, so when you create a rule from source A to destination B you don’t have to create the B to A restriction.
Then you go to the permissions page and assign the new role created to the users or group you`d like to have this role.
About the other cases:
Add a new project restriction to the role you have created > Task supression is not allowed
Users on this role will still be able to open and close tasks, you can add another project restriction, “closing or opening task is not allowed” to prevent them to close tasks.
I’ve never seen this restriction, I don’t think is possible.
This custom roles restrictions may behave not the way you’d like if you add too many of them. This ones I mention here, drag and drop, supression and open and close work well.