If you need to move thousands of Azure blobs, or very large ones (many GB) then you can either use AzCopy (a command line tool which you run locally), or if you want to do things programmatically, you can use the Microsoft Azure Storage Data Movement library.  That library is apparently what AzCopy uses behind the scenes.

(For some reason, Microsoft don't link to the data movement library from their page about AzCopy - they should do).

See here for AzCopy and here for the Data Movement library.