Ok. Tsunade was basically out of chakra and has just enough to save the other Kage, she is also the most critically injured out of all of them and the only one on scene that can revive/heal them. Last time we saw them, she was going to sacrifice her remaining chakra (her life as well) to heal the others since she was crushed from the waist down. It's set up for her to die. Anyone still denying this needs to look at these facts for a few hours and just deal with the fact that she may very well die in this arc.
Now for the OP.
Minato is the fastest of the Hokage.. that and the fact that his motivation to get there faster is his son. So of course he is there before the others. I'm guessing we will see them come in, I think Hashi will see Tsunade and Tobirama will try to do something... but it will be in vain, so Hashi will storm the battlefield and that will be his motivation to bring down Madara for good... or at least that's what I hope happens.