Holacracy Community of Practice Archive, 2015-2019 Community Holacracy Web Site
Hi Anja, yes processes are rarely “lived” when they are detached into separated description documents buried into some company folder. There are some modern  ways to overcome that, i.e. by adopting business process management suites by which your personal involvement in a process is triggered by some IT platform that keeps track of what is expected from you in which step, therefore implying that you have an implicit “role” in that process. 
If by Lean Management, in production environment, we essentially define a set of practices aimed at increasing process effectiveness by continuous improvement, then we could say that the way Holacracy targets that aspect, is by basically removing the (human) process from the document and spreading it into role accountabilities. Each process participant can then look at his role description(s) and find out what is expected from him as far as inputs/ouputs/work to be done/interactions with other roles, in each specific process are concerned.
The advantage is twofold:
1) you, as a role filler,can look at your roles an know exactly what is expected that you do
2) you can bring your tensions relevant to the effectiveness of the process by proposing changes to various role dimensions ( accountabilities and domains: not only yours but those of others as well), the result being exactly what lean management aims at.
There are other ways you could encode formal processes into an organization adopting holacracy (by means of policies the refer to specific constrains and rules of engagement you want to enforce, or  even by pointing out that a specific process must run according to the way it is designed by a specific process designer role on a specific platform… The possibilities are virtually endless). Let me know if this response targets your need.