I'm not sure what you're asking. Most live distros have a way of installing on the device. That wouldn't change what's on the USB. You'd end up with a device with Linux installed, and a bootable USB drive that you could still use to...do something I guess.

Also not sure what you're trying to do.

You could technically install your Linux distro to the USB first and then clone your image to another system. Wouldn't recommend since that other system is likely to be different (cpu, gpu, etc.). This is also a lot more advanced since you would likely have to use something like dd or something else.

I think the more common thing to do would be to flash the LIVE image to the USB and then boot from it on the target machine to then run the installer from the LIVE image. This shouldn't affect the USB if you do it right.

Yeah? Installing it doesn't move it from the stick to the PC; it makes a copy.

Try maybe ? https://www.ventoy.net/en/index.html

So you’re booting from what is on the USB stick, it’s acting as a little hard drive that your machine is loading the OS from. Then the is running on the system installs it’s self to the actual hard drive. That installer build of the OS is still on the USB stick. What is on the hard drive might be different as during installation and set up, it may have been configured differently. But the USB stick should be unchanged.

im not sure what you are asking honestly but there are distros that run from memory so you can boot from an iso but get better performance. Of course this eats ram but they tend to be light distros and with modern ram sizes it works out. puppy is one.