Fine, does that come with a right to define the immigration process they will use?

I mean, somewhat? They have a right to have their cases decided by an immigration law judge, with all the attendant rights that come with that (right to receive notice of the hearing, right to provide evidence to support their claim, etc.). And they have a right to have their case decided on the legal standards in the statute. And of course, all of the things that generally apply to how the government treats people would apply to them - the government can't steal their stuff as part of the process, for example.

But they don't have a right to move freely about the country, if that's what you're asking. They can always just be detained. But while the Administration has the legal authority to detain asylees pending their hearing (except for kids), they don't have the resources to do it. The wait time is too long, and there's too many of them. They don't have the budget to do that.