Hi! The buffs like breakthrough passive for Hashirama [First Hokage] will remain on the field the entire battle, regardless if the ninja dies in battle. Since it is permanent, it cannot be removed.

Jib Edit: I have edited this post after confirming with the team as well as testing the Hashirama [First Hokage]'s Ninja God Passive after Skill Breakthrough. Sorry for any confusion.

Courtesy to RenjiAsuka and Ox-BADTEMP for providing the information.

Hi! I have just tested it as well as asked the team for confirmation and it looks like I was mistaken. Sorry for any confusion.

I have also updated the previous post to avoid any confusion.

Iruka's buff and barrier will remain for X rounds even if Iruka dies. However, Iruka's buff is still susceptible to ninjas who remove buff and/or shields.

In regards to your first question. Yes, Hashirama [First Hokage] with Ninja God passive skill broken will have the buff persist even when Hashirama [First Hokage] dies.

You are correct in regards to Asuma [Wind Blade]. I have confirmed that Asuma [Wind Blade]'s "Pose of Invincibility" passive is a conditional passive and requires Asuma [Wind Blade] to be on the field (alive/active) as well as lineup ninjas meeting a specific hp requirement for the passive to take effect.