To answer your main question, you can transfer CRM databases from single box to multiple box environments, the key thing is the CRM database. Note that just restoring will not work, you need to also run the CRM Deployment manager to import the CRM database to a new CRM system, it will upgrade if its one version behind or just import if it isn't. You cannot "downgrade" or import databases older than 1 version.
From what I am understanding you have a full CRM 2013 Environment with 4 boxes
You ultimate goal is to have a CRM 2016 environment (4 boxes).
I highly suggest you research/test the process and perhaps get some experienced help. It will be worth it.
From a very high level and from what little I know of your environment, here is how I would approach this:
Set up a temporary system with everything (SQL, CRM 2015, SSRS, etc). Since this is not going to be for production, you don't need to split everything out. This is just for your "hop" from CRM 2013 to CRM 2016. VMware or Hyper-V would work great for this.
Back up your CRM 2013 org database (<<orgname_MSCRM>>) (in SQL).
Restore the org database on your CRM 2015 temp environment (in SQL).
Run the CRM 2015 Deployment manager that will both import and upgrade the database into CRM 2015.
Make sure it works in CRM 2015.
Back up your CRM 2015 org database (the one that just got upgraded) (in SQL)
Prepare a new CRM 2016 environment (4 boxes if that works for you).
Restore the CRM 2015 org database in the new CRM 2016 environment.
Run CRM 2016 Deployment manager to import and upgrade the database.
Fix any scripts, reports, etc.
You should be good to go. Retire your old CRM 2013.
Like I said, research and ideally run through the whole process on a test run, noting steps, issues, etc. Before you "go live". This will be a significant multi-step project and should be treated as such.
I can't stress enough to read the Microsoft documentation!
Cheers
Nick
