Since you have to update site column used in Content Type, Field.UpdateAndPushChanges method could be utilized. It commits changed properties of the field and propagates changes to all lists that use the field.
The following example demonstrates how to update field Description
and propagate this change to all lists:
$ctx = New-Object Microsoft.SharePoint.Client.ClientContext($webUrl)
$ctx.Credentials = New-Object Microsoft.SharePoint.Client.SharePointOnlineCredentials($username, $securedPassword)
$field = $ctx.Site.RootWeb.Fields.GetByInternalNameOrTitle($fieldTitle)
$field.Description = $fieldDesc
$field.UpdateAndPushChanges($True)
$ctx.ExecuteQuery()
Note: The specified script have to be executed per every site collection.