As stated in the title, I was having issues with cookies of kibana instances behind a load balancer and even after taking into consideration what's written on the page concerning Kibana in production, it didn't work.
I found the solution on the following topic:
Random Logout of Multiple Load Balanced Kibana on Multiple Elasticsearch Clusters
Not being able to answer the proper topic, I just created this one to say that the idea of @ea1987 to have this behavior documented inside the above-mentioned page is very relevant and would make a lot of sense.
Let me know if it's not the right place for this proposition.