If your boss is disorganized and a procrastinator, then make it as easy as possible for him. If there are forms, fill them out for him so he just has to sign. Make him feel comfortable with the transition. Procrastinators are overwhelmed and anxious. Go to him in person to sign whatever papers you need. Make it easily digestible for him. That’s how you manage up for someone like that.

Is this different role under a different supervisor? If so, then you don't need his permission.

Find a different job.

You need your first identity what you mean by support. He sounds like a blowhard who will say he supports you but not do anything. You need to define clearly if what you need is a letter from him or for him to talk to x about your future or what, and then be on top of it, as in write the letter yourself, or make an appointment with X and your boss and have the talking points ready and don't expect anything from boss but nodding along.

Get your boss to write. it. down.

My boss doesn’t take responsibility for what he needs to do and doesn’t follow through with his promises. He told my coworker he’d give her time off if she worked extra on a project for him. She took the project and worked late many days, only to be denied time off when she requested it.

Another coworker was sexually harassed, reported it and my boss did nothing. HR eventually got on him about it, but it was unnecessarily distressing for my coworker.

I would like to take on a different role, and my boss said he’d support me in it. I can’t do it without his permission and support. Any tips on how I can help him follow through with his promise to support me in this?