Is it permissible to transfer to another department within the same institution through an influential person, if the work itself is in different circumstances and harms no one and benefits the one transferring?

There is no harm in moving to the new department, even if it is through the intercession of a person of influence within the institution, as long as it does not wrong anyone or infringe upon their rights. Allah Almighty said: "Whoever intercedes with a good intercession will have a share from it, and whoever intercedes with an evil intercession will have a portion from it. And Allah is over all things a Maintainer."