Recovering a Mac Pro 5.4

The Mac Pro 5.1 (2010-2012) cannot upgrade to macOS Catalina, see below.

Check compatibility

If you’re using one of these computers with OS X Mavericks or later,* you can install macOS Catalina. Your Mac also needs at least 4GB of memory and 12.5GB of available storage space, or up to 18.5GB of storage space when upgrading from OS X Yosemite or earlier.

MacBook introduced in 2015 or later

MacBook Air introduced in 2012 or later

MacBook Pro introduced in 2012 or later

Mac mini introduced in 2012 or later

iMac introduced in 2012 or later

iMac Pro (all models)

Mac Pro introduced in 2013 or later

There is no Mac Pro 5,4 model. I can only find MacPro5,1 which is a Mid-2010 model. To find the complete model information select About This Mac from the Apple menu. You can create a snapshot of the dialog as follows:

How to take a screenshot on your Mac

Capture, Save or Record Screenshots in Mac OS X - Help Desk Geek

OS X Screen Capture Shortcuts

(⌘⬆︎3) Command-Shift-3 Capture the screen to a file

(⌘⬆︎⌃3) Command-Shift-Control-3 Capture the screen to the Clipboard

(⌘⬆︎4) Command-Shift-4 Capture a selection to a file

(⌘⬆︎⌃4) Command-Shift-Control-4 Capture a selection to the Clipboard

(⌘⬆︎5) Command-Shift-5 Invoke new screen grabber

You can then post the image in the forum's message editor.

The Early-2009 model, MacPro4,1, came with Snow Leopard 10.6 installed and can be upgraded to El Capitan. The 2010 model came with Snow Leopard 10.6.4 installed and can be upgraded to Catalina (if you install a Metal capable GPU.)

You will only be able to install Snow Leopard with the Software Install disks that came with the computer, although the 2009 model could be booted from the last retail version of Snow Leopard - 10.6.3. Both need to be updated to 10.6.8 in order to access the App Store and download macOS upgrades.

The installation window that you saw with the Yosemite and El Capitan downloads is not installing the OS.

An OS is about 30GBs in size and takes about 30 minutes to install. If you read the window prompts properly they say this will take 7MBs of disk space. What is happening is the InstallMacOSX.pkg is being converted to either Install OS X Yosemite or Install OS X El Capitan.app which you can find in your Applications folder, have a look.

To start the installation of the OS you need to double-click the Install OS X Yosemite.app or the Install OS X El Capitan.app

each should be about 6GBs in size. These installers will self delete after you have completed the installation process.

If you want to keep a copy of these installers make a copy of them somewhere else on your mac or on an external drive.

You can also make a bootable USB installer of the El Capitan installer using the instructions here,

How to create a bootable installer for macOS – Apple Support

If you want to make a bootable installer of the Yosemite installer then use the appropriate instructions for Yosemite here,

https://diskmakerx.com/download/

As mentioned there is no Mac Pro 5.4. Just to make sure the Mac Pro of 2009 vintage is a large silver tower desktop machine.

The Mac Pro from 2009 is a 4.1 and can only upgrade as far as Mac OS X El Capitan.

The Mac Pro from 2010-2012 is a 5.1 and can be upgraded to macOS High Sierra, and with a 'Metal' capable graphics

card can upgrade to macOS Mojave.

If it is a Mac Pro 5.1 (2010-2012) read the instructions in each of these articles.

How to upgrade to macOS High Sierra - Apple Support

How to upgrade to macOS Mojave - Apple Support

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.