They eventually go away. As a former mission runner, it’s important to allow them to remain open. Let me explain from my point of view.
A bomber is easy to pop. There are many, many pvp-fit pilots that sit in systems waiting for mission runners to pop a mission so they can run to it and get easy kills. Some pilots spend their whole day doing nothing but waiting for them to pop open.
As a result, many mission runners will come into system, pop the mission, then go to the next system and do the same. Then, when all popped, come back and take a shot at the missions. It’s safer that way. If I went into a system and nobody was there, I’d pop the mission and immediately go to it and complete it. However, there were some systems that I knew a guy in it was only waiting for mission runners. In those systems I always popped them and left, to come back later.
tldr; Sorry, but it’s safer for mission runners.