KB: Transfer workspace data

Solution Details

If for the purpose of data backup, it is done daily on our end automatically, though data can be downloaded manually at your own discretion:
Frequently Asked Questions

The specifics of how data can be extracted offline can be found at:
Extracting Data from a Workspace

Projects:
If you want to "migrate" your design data to another Workspace, there is no automated workflow, but you can take the projects offline (within Altium Designer, Project » Project Packager - be sure to check the "Unlink project from the server during packaging" check box in the second screen):
Working with the Project Packager

You can then make it available online again in another workspace. This process will not retain the project history, and if there are components managed in workspaces used, you would need to either migrate corresponding components into the new workspace or re-create them off of cached copies within the design, as these projects no longer have access to the old workspace. Once that's done, cached components instantiated in the migrated design can be updated from the new workspace/libraries.

You can use Find Similar Objects to select all the components in the project and then change the source library in the Properties panel by selecting a component, right-clicking on it to choose Find Similar Objects. In the Find Similar Objects dialog, set Object Kind-Part to Same, enable Select Matching and set to Project Document and click OK. In the Properties panel, change the Source Library which opens the Component Source with the option to keep the current Design Item ID or reset the Design Item ID and apply the change after choosing an option.

Here's documentation on Using Find Similar Objects:
Using Find Similar Objects Tools

Though not a native scheme to manage projects in Altium 365, it is also possible to use an external VCS to store a workspace project:
Using External Version Control
Making an Existing Project Available in the Workspace

Components:
To transfer components from one workspace to another, you can use the content cart:
Acquiring Workspace Data Using the Content Cart

The direct transfer using content cart may cause some issues involving, say, component templates, which is available on a workspace with Pro subscription but not with Standard subscription. 
In such case, components can be taken offline in a form of file-based Intlib:
Using the Content Cart for Acquisition to an IntLib

And imported back into another workspace:
Importing Existing Libraries to Your Connected Workspace

Part choices associated with components may need to be added again if this information is required.
https://bugcrunch.live.altium.com/#/bug/20809

To extract component parameters solely, Export Grid command in Explorer panel may come in handy:
Export of Grid Data

A caveat here is that there is a limitation of 10K row items inclusive of various revisions, per a given grid export.  Such manual operation is not recommended as a part of regular workflow, in favor of automated/periodic synchronization:
Component Database to Workspace Data Synchronization (uni-directional, csv/xls/ODBC, requires Pro subscription and above)
PLM Integration (bi-directional, PLM, requires Enterprise subscription)

Merely to refer data to users without an access to Altium Designer client (i.e. procurement), Library Health Dashboard for components is a more convenient interface reflecting real-time status: 
Library Health Dashboard