A statement from the company said the files that were stored would be automatically transferred to Nextcloud, while the import of calendar, contacts and other data was available in some cases.
Nextcloud cited the rise of violations of privacy and disruptive changes by proprietary services like Google Photos as the reason behind its move.
"Unfortunately the barrier to self-hosting services is steep for many non-technical users," the company said. "Nextcloud has been on the forefront of lowering this barrier, with its unique Simple Signup which provides users with a 5GB free storage Nextcloud account at a local, reliable hosting provider in under 30 seconds.
Jan-Christoph Borchardt, design lead at Nextcloud, said: "To really change the dynamics of the Internet and the massive privacy violations that have become core to the business model of many Internet giants, alternatives have to challenge them in capabilities.
"Migration has to be seamless before the vast majority of users is able to move and adopt privacy-friendly solutions and today, we make a big step forward in that regard."
Users could migrate from Google and calendars, contacts, photos, files from Drive and documents from Google Docs would be migrated into their respective Nextcloud services, the company said.
Users of Google Photos were recommended Nextcloud apps like Maps and Facerecognition as replacements.
For Dropbox, file import was available, preserving directory structure. Microsoft OneDrive users could also migrate file and folder structure, with other data migration options on the Nextcloud's roadmap.
Andreas Rode, chief commercial officer at Nextcloud, said: "These migration services are a great win for the data sovereignty of consumers all over the world. Millions of users have access to Nextcloud servers and we look forward to many more to sign up at a provider or set up their own server.
"We expect many hosting providers do enable these apps for their users, easing their way into a private, secure alternative."