If you want to migrate data from site collection to site collection. You can use different methods.

1. Explorer migration.
Simply open the explorer view on both pages and copy past the data.

2. Sync all the libraries where you want to put data in.

Simply open the site go to the document library and press synchronize. Once you have done this you can find the sites in your SharePoint favorite in the Explorer on your Computer.

I know there are lots more methods to perform a content migration.

ShareGate

What is Sharegate.

*From the sharegate website
Sharegate is built to migrate everything inside out of SharePoint & Office 365. Migrate entire site collections or decide what you want to move: sites, libraries, lists, workflows, Nintex workflows & forms, documents, and much more.

With Sharegate it is really easy to migrate data between site collections and different Office365 tenants.

When you start Sharegate you need to setup the main Office365 tenant to start using sharegate. Once you have done this it wil look like this. (i used my Microsoft live account for my experience)

After when you have connected to you tenant you want to start the migration of content. for this option you choose migration.

You will be redirected to a page with the following notification.

Click Launch Migration to open the Sharegate migration app.
From this point you will be able to navigate thru the site collections or tenants you have add and you can start migrating data between the locations.

With this tool you can migrate whatever you want.

• Sites
• Documents
• Libraries
• You can import data from a fileshare into SharePoint or Onedrive
• And you can even Export data from SharePoint or Onedrive to a fileShare

Sharegate has lots of more functions you can check them all out at The Sharegate website. You can also get a free Trial for 15 days.