The ISS will remain at a relatively constant magnitude as it passes over and will diminish as it closes on the horizon, thought this will depend on the level of light pollution.

Likewise most satellites will remain at a constant, though lower magnitude than the ISS, the Iridium satellites however due to their construction give the exact light display that has been described and has already been mentioned a few times on here, due to the panels they act as large mirrors and will catch the sun and give the illusion of a flare, hence the name and these do indeed travel in the direction mentioned.