After deploying the new SharePoint 2013, I have migrated all my content from SharePoint 2010 using the database upgrade process.

To do so, I backed up the content database from my previous installation, running on SharePoint 2010, and then restore it to the new SQL server.
Then, after creating all the web application, I attached the content databases using the PowerShell New-SPContentDatabase -Name <ContentDbName> -WebApplication <WebApplicationName> to upgrade the content database to the new version.

Once content databases have been attached to SharePoint 2013 web application, the first time a Site Administrator connects to the site, he gets a message to propose to upgrade to the new version

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At this stage, you can postpone (Remind me later) or start the upgrade process (Start now)

Hitting the Start Now link, sends you to the Start Upgrade page

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At this stage, you can start immediately the upgrade process (hit Upgrade this site collection) or perform a staged upgrade (Try a demo upgrade)

If you hit Try a demo upgrade, you will be asked for creating a site collection test upgrade

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You will receive an email to remind you that a demo upgrade request is in progress

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If you hit the Upgrade this site collection button you will be asked to confirm you are ready for upgrade: hitting I’m ready will start the upgrade process while hitting Let’s hold off will take you back to the upgrade page

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While the upgrade process is in progress, you can continue to use your SharePoint site; you will have a banner to inform you that an upgrade process is in progress

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