I am in the process of planning a SharePoint 2010 upgrade to SharePoint 2013 and then to SharePoint 2016. Both SharePoint 2013 and SharePoint 2016 farms are built and include Managed Meta Data Service, Secure Store Service, Business Connectivity Service, User Profile Service and Search Service. How does having these services already installed on my SharePoint 2013 and SharePoint 2016 farms affect upgrading these services from SharePoint 2010? Can I upgrade these services if they are already installed on the SharePoint 2013 and 2016 farms?
You should follow the Microsoft guidance and recreate the Service Apps with the old databases so that they will be upgraded. I always start with the Managed Metadata Service since the User Profile database, content databases, and others rely on the MMS identities for their values.
For the Secure Store you will need the passphrase so that you can decrypt the credentials.
You didn't mention it, but the Search Service Application has a special comdlet for the upgrade process so that you can preserve your Managed Properties.
If you have a Pluralsight membership I cover all this in my course on upgrading SharePoint. It also includes all the scripts you need.
Based on Microsoft' article at here, once old databases are restored in the new SharePoint Farms, the Service application databases are upgraded during creation of Service Applications in the new Farms.
When we upgraded from SharePoint 2010 to 2013, we restored the old Managed Metadata Service Application Database in the new farm and created the Managed Metadata service application by providing the database name as shown below
Add-PSSnapin Microsoft.SharePoint.Powershell $mms = New-SPMetadataServiceApplication -Name "Managed Metadata Services" -ApplicationPool "SharePoint XXXXX Service Pool" -DatabaseName "MetaDataXXXX" New-SPMetadataServiceApplicationProxy -Name 'Managed Metadata Services Proxy' -ServiceApplication $mms -DefaultProxyGroup Set-SPMetadataServiceApplication -Identity 'Managed Metadata Services' -DatabaseName 'MetaDataXXXX'