The simplest answer is: it depends!  You need to understand the goals of the business, the current organizational structure, any constraints, etc.

Example:  Should it done by the same group? 

• If the category change requires specialized skills that only certain workers have then the business process would need to route the issue to the specialized group.  In this case, the needed skills would be a constraint.
• If the business has a requirement for single point of contact/handling then you may have to keep the issue with the person who was originally assigned the task.
• If there are not constratins then you would want to pick the option what his most efficient (takes late time) and costs less.

Example:   Should it starts from the beginning even if some time passed or should just continue and calculate new time for resolving that issue?

• If what's important to the business is the total duration needed to resolve the issue then the clock would continue (without reset) since from a customer's pespective it doesn't matter how many times the issue changed hands but only how long did it take to resolve it from when it was first reported.
• Having said that, if the business wants to keep track of how long a task is in each worker's hands then you may need to reset the clock when the issue is reassigned.
• You could do both aka track the total issue duration from begining to the end, at the same time, have separte records for how long each assigned person worked on it.

The bottom line is that there is not right or wrong answer in the absence of requirements.  The questions you need to ask once you have a solution are:

• Does the solution satisfy all buiness requirements?
• Does the solution conform to any constraints (legal, market, regulatory, etc.)?
• Is the solution as efficient as it can be?

If no, then you can go back to the drawing board and rething the solution.

Hope this helps!

Adrian