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The easy method.
http://www.macrium.com/reflectfree.aspx
 
Macrium Reflect v6.1 has a WinPE Rescue Media Wizard, that can create a specific version of the WinPE media for each version of Windows (Win 7/8/8.1/10).  This media needs to be created for each PC locally.
 
If you lose your Windows operating system, you can start your PC using Macrium Reflect rescue media on CD,
DVD, or USB stick. This makes creating rescue media the first thing you need to do with Macrium Reflect. It contains a bootable, lightweight version of Windows and a full version of Macrium Reflect.

This lightweight version of Windows is called Windows Pre-installation Environment (also known as Windows PE or WinPE) and is provided by Microsoft. When you create rescue media, Macrium Reflect downloads Windows PE
automatically for you and writes it to your media. It downloads just those components you need to rescue your
system.

Windows PE and the rescue environment

Windows PE is a reduced version of Microsoft Windows that is designed to boot from CD, DVD or USB on a wide
range of hardware. When you run the rescue media wizard, Macrium Reflect automatically downloads the Windows
PE components from Microsoft and builds the rescue environment locally.

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Caa2
Yes, Thank you. I have made a USB stick of Macrium Reflect just as you suggest.
 
The recommendation to try building a MultiBoot rescue stick has taken my interest further.
 

I was following this thread, and after re-reading your OP, I thought you originally just wanted a recovery boot with WinPE.  The Macrium process is plug and play, does it all for you.
 
I am curious now, what you are trying to accomplish? 
 
There are many tools available for PC tech, depending on what you want to do.  I build Windows and Linux systems and have several tools at my disposal for online and offline imaging and bare metal recovery, disk partitioning, file recovery, malware removal, etc.  Some are easier than others to deal with.  It's like a toolbox.  I like a tool for each task, no one size fits all.
 
Bottom line is, simpler is usually safer.

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Caa2
Hi, 
 Yes, the original post was specifically asking about making a WinPE based USB stick for a single application, but after considering the ideas suggested by arachnaut I have become interested in the idea of making a custom multiboot rescue package.
 The question was answered via explanations provided by slartabartfast and arachnaut.
 
 I can make several individual USB sticks with a single application on each but the multiboot idea seems like a good option.
 Finally, I am enjoying learning about the multiboot concept and I would like to see the idea through and get something working on my system.
 
 Some of my older systems have legacy BIOS options, but I would like to find a solution that works for my most recently assembled computers as well.
 

It's great to hear that your original question was answered, and that your thirst for knowledge continues unquenched.
 
You received some great advice from some smart guys. It appears that you took it all in and soaked it up
 
Regarding the choices in reliable imaging solutions, I am also re-evaluating what I am using currently.  I am having a look now at Macrium and Paragon. 
 
My current tool, the native Windows Image utility, has been with me since Windows 7.  I have confidence so far that it can do the job reliably, and with the Windows recovery disk, I can do a complete disk restore anytime it is needed.  Since I moved to Windows 10 in July, I defaulted to the native image because I am now running UEFI BIOS and a GPT partition on my boot drive.  I wanted to make sure the built-in tools could cover those changes, and as far as I can tell, they are.  I have the Windows recovery tool on a CD-R and a USB stick, & can boot and access my images OK.  I can also mount the VHD image as a drive letter in Windows, then browse files and folders and copy files directly from the image if I wish.
 
So why am I considering the other solutions if Windows is doing what I need?  Well the file management, for one thing. It is hands on and requires me to rename & or retire the old images manually to free up space.
 
I am hoping that the WinPE recovery media with Macrium or Paragon is as solid as the Windows 10 native tools, and I would hope so, since the WinPE originates with Microsoft.
 
Bottom line, full disk imaging is the only way to go.  I had a few bad experiences with file & folder backup schemes years ago.  Once I tried a utility called Drive Image by Image Quest I was hooked, and stayed with that while using Windows 2000 & XP.  They were eventually bought out by Symantec and their code was merged into Norton Ghost. I couldn't believe how easy it really was, and how much confidence you gain by knowing that you can reset your PC back to any image you wish.
 
Good luck on your imaging plan!

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Caa2
 
 The moral of the story is that if I had created the WinPE rescue media on the actual hardware that I wanted to use the rescue media on then the .wim boot image would have had the exact and correct drivers. When I made the WinPE .wim boot image on another computer using differing hardware I ended up with a less than ideal situation.
 
 One circumstance was going to take 44 hours to complete the restore while the other circumstance completed it in 11 minutes.
 
 Info worth knowing??? :-)

 
Yup!  Good to know.  The school of life, huh?
 
I didn't notice anywhere that it said you must use a rescue disk created on specific hardware, but the description provided in the Macrium user guide implies that may be the best option...
 
What version of Windows PE should I choose?
You should choose a version of Windows PE that can access your System drive and also your backup
location. The default option selects the Windows PE version that is the best match for your Windows
operating system. This enables the rescue media wizard to automatically copy any required drivers for
Network, USB or SATA controllers. However, versions of Windows PE that are more recent than your
Windows OS may already contain compatible drivers and also offer additional support for USB 3.0.
 
Windows PE (WinPE) is packaged with a large collection of drivers but there are many devices
that are not part of the WinPE driver package. If your device is not compatible then you must add
its driver so WinPE recognizes it and communicates with your device.

The wizard checks whether your device requires drivers adding to WinPE. It builds a list of devices in your
computer that are either Hard Drive/RAID controllers, Network Interface Cards, USB controllers or USB hubs.

For each of these devices it checks if:
The device is supported by default in WinPE
There is a compatible driver in the host operating system
There is a compatible driver already present in the collection of drivers on previously created rescue
media
 
Supported devices have a status of either:
Device Support in WinPE
Compatible Device Support in WinPE
Copy Host Driver
Driver already present in Drivers folder

Even if a device is supported you can choose to update it and use a different driver, you do not need to use
the driver provided by WinPE. Being able to update drivers in this way is useful if you experiencing issues
with performing backups or restores in WinPE, for example, if restoring runs slowly from a USB device.

You can find driver install packages in several places. Most manufacturers create a driver folder on the hard
drive of a new PC, often named after the manufacturer. If this is not present, you can download drivers from
the device or PC vendor's website. For older PCs, manufacturers of devices and motherboards included
driver CDs in the packaging, however, these may not be the latest.

Click the 'Update Driver' button to search for a compatible driver.
This presents a wizard. This wizard is straightforward to use, the first page prompts you for a folder to scan
for drivers and whether you want to include all sub-folders.

Yes, it would not make any difference on which system you created a WinPE iso. WinPE does not take its drivers from that system, but only has the drivers it ships with. You would still need to add those machine specific drivers to your flash drive version when you set it up, even if you set it up on the machine that required those specific drivers. The only advantage to setting it up on your target machine is that you might be able to find the necessary driver stored there, but in the general case you would need to do what you did, i.e. find the appropriate driver files somewhere and add them to WinPE manually.
 
This requirement for the imaging system to be able to load the necessary machine specific driver is not unique to Macrium. When saving an image in most imaging programs you are presented with a dialog to save or load those drivers before saving the image. Otherwise you can have a good image of you machine, but in the pre-restore environment, which is not the same as the environment saved in the image, there may be no way to communicate with the machine to do the restore.
 
This is the same issue that occurs when you are installing Windows from the preinstallation environment using an install disc or iso. If you do not pause the installation to install needed drivers, the installation will not succeed.