Follow-up or "chain" missions are, as noted above, generally intended to be extra rewards for a job well done - a tiny allocation of cargo to move for several million credits, or a nice juicy pirate to assassinate. The high value of the rewards also means they're a good source for max-rep, max-inf or max-mats rewards options. If you don't take the missions, they eventually disappear (I think they stay up for a week?). They won't get upset with you if you choose to ignore them.
The missions are as semi-persistent as regular missions, but If you don't like your follow-on mission and want to re-roll the dice, you can: if the faction that issued it has multiple stations, visit another station they own. The other stations will generate their own follow-up mission list. The fluff-line in the comms message about the follow-up mission being offered at a certain station is lying; any old station the faction owns will do.