Background: I was offered a PhD position by the advisor of my current masters thesis and also by the corresponding professor at the local university. Now that i thought more thoroughly about pursuing an academic career i want to apply to a different university to gain experience abroad (eg. Switzerland).
Obviously my advisor and the professor are interested in my work, is there any chance they will support me with a letter of recommendation? Or more generally if the professor likes your work, why would he or she support you to apply somewhere else?
Edit: To expand on the answers, will i risk my chance of PhD at the local university just by asking for a recommendation (e.g. EPFL seems quite competetive)?