If you create a SharePoint site column (a note field in this case), associate it with a site content type, and then associate that content type with a list in a sub site, the site column will be available on that library. Obviously right?
However, when you update the site column (and push all changes to lists and libraries) not *all* of the changes you make are in fact pushed down. An example of this is the setting that dictates whether a note field should allow rich text or enforce plain text. If you change this setting at the site column level it will *not* propagate to libraries which already exist. New instances of the column (say if you associated the content type with a list for the first time) will be configured correctly, but existing list-level instances are not updated. NOTE: This is only true for properties specific to particular column type; common properties such as ‘required’ will be pushed down to existing instances of the column at the list level.
So you want to change a list-level instance of a plain text note column to a rich text note column (or vice-versa, or otherwise change column specific properties or another field type)? You need to do it for every list where the column is in use. That would be very tedious to do via the SharePoint UI, but you can’t anyway. The UI only supports changing the set of common field properties (type, required, hidden, etc).
In comes PowerShell. Below you will find a script which updates a plain text note column to be a rich text note column. It is important to note that this script only updates the list-level columns and not the site column. This means that after running the script, new instances will continue to inherit the site column configuration.
The script takes advantage of recursion using delegate functions which is an approach I blogged about here: PowerShell Recursion with Delegate Functions
Credit also to Chris O’Brien’s topofscript.ps1 for the CSOM integration bit: Using CSOM in PowerShell scripts with Office 365
The script is written for SharePoint Online (and assumes that the SharePoint Online Client Components SDK is installed) but for this to work on-premises you would only need to update the referenced assemblies (v15 for 2013) and modify the code which passes the credentials.