How to install Windows 10 ONTO a USB flashdrive

"so that I can just plug it in to a pc "

For multiple reasons, you probably don't want to do this.

1. Licensing. You wish to be able to plug this USB into different random PC's and have 'your system' as it is?
Licensing will kill this.

2. Speed. Most systems will not boot from a USB 3.0 port. USB 2.0 will be supremely slow

3. 32GB. That is just barely enough. You will run into issues with updates, etc.

4. Win 10 Enterprise can do this easily, with Win2Go. But you don't have a valid license for Enterprise.
Other methods can work, but very kludgey.

What is the basis for wanting to do this?

A Linux Live CD (or USB) will do this.
Windows, not so much.

Guess it's time to learn Linux!

you cant run your windows on any one computer , you will have to deal with different drivers and conflicts and BSOD ...

Forget about it . even if you could do it , will not work on different hardware .

Accidentally trashed the Win 10 installation on my UEFI yoga pro 3 when I tried to move $USER to another partition on my SSD. Luckily I am dual booting with Linux and could at least try and repair Win through that OS. I struggled for ages to reinstall Win 10 to SSD without destroying my Linux OS but no go, Win wants to hog the whole drive, a least on Lenovo UEFI machines with SSD.

Eventually gave up trying to install Win to SSD (until I get another SSD to clone as back up) and focused on finding a USB solution.

Explored other 'straight' options to install Win to USB, including win to go. Nada. At least on my machine.

I had previously managed to install non-UEFI capable versions of Linux to usb by creating a VM in Virtualbox in Windows (by setting up the VM with no HDD, an optical disk containing the installation ISO and a USB stick), install Linux to the stick, install GRUB. Shut down the VM. Shut down Windows. Fiddle with BIOS to boot from the USB and voila! A functioning Linux OS that was never intended to be UEFI bootable! (In fact I then cloned the Linux OS to an SSD partition, updated GRUB and that is how a non-UEFI Linux distro ended up fully functional on my UEFI machine. Thank you GRUB).

I tried the same trick in reverse with Win 10, starting with a VM in Linux installing to USB, but, as with all things Microsoft, it didn't want to play.

How is this relevant? Well, having been put though the mill and become familiar with VM's by force (as it were) I am currently happily running both Win 10 and Win 7 as VM's through Virtualbox hosted on my Linux system. Each Win Virtual Machine is installed with its virtual drive on a separate USB 3.0 stick.

Since the VM's run on mainly generic hardware then they can boot from other hosts running Virtualbox. Not a solution for high performance video processing or hardware dependent stuff but for the declared reason in this thread of portability, sound as a pound. And, of course, because the VM's are sandboxed, there is no chance of someone claiming that a virus from my VM trashed their host.

Perhaps not quite perfect but as close as I have got....... so far.... unless someone knows better?