First, the required profile needs to be present. LR 5.4 supports the X-T1 in terms of understanding its Raw format, but will not come with a profile for every lens which can be mounted onto that.
A profile is made for each specific lens, as mounted on a certain camera. It can also be used with pictures taken on a different (reasonably similar) camera body than the one originally used during profiling. This is subject to the sensor format: you can employ a given lens profile made for full frame, on an image from a crop sensor behind the same lens - but not vice versa.
Next, the image type needs to correspond to the image type which has been built into the profile: Raw, or non-Raw.
Lightroom offers only profiles of the right type for the image(s) concerned. There are many more provided for correcting camera Raw, than for correcting camera JPG. That is also the case with additional profiles, e.g. those accessed via the Adobe lens profile downloader utility.
Finally, in the event that there simply is no profile out there for a given lens, it is not particularly difficult (though a little tedious) for the end user to generate one, using a further Adobe utility which is downloadable as a package, including a range of checkerboard calibration targets, each of which needs to be printed at a stated physical size.
Thank you for your response. I am, in fact working with Raw images (converted to DNG in the import process) and I realize that the sensor characteristics of the 2 cameras are similar. But I would have thought that any pin cushion or barrel distortion characteristics of each lens would be unique. My images, for instance, have been shot with the Fuji 23mm lens and the 18-55mm lens whereas the offered x100 profile is, I believe, a fixed 35mm lens. So wouldn't I be better off using no profile and manually correcting the images (or creating the custom profiles you mentioned) than just applying the X100 profile to images taken with totally different lenses?
I sent this, earlier via e-mail, and it hasn't come through:
The mirrorless cameras usually have built-in, always-on profiles, nowadays, both in camera and in Adobe software, so there shouldn't be any reason to use another camera's profiles, from back when Adobe didn't build them in.
Photoshop's ACR plug-in will actually tell you when a built-in profile is being used, but so far LR doesn't, maybe LR 6 will?
If you don't have Photoshop, if you want to upload a raw file I, or someone, can check it for you, to see what lens profile corrections are occurring if any. To upload a raw file, put it on somewhere like www.dropbox.com and post a public download/share link to it.
Thanks - yes I have PS so I'll check Camera Raw. I was wondering about whether there was redundancy between the in-camera correction and the LR correction but I thought the in-camera only applied to jpegs. I will check it out. Thanks again.
In camera correction applies to JPGs but the camera also applies it to the LCD/EVF version of the image in realtime so the end-user never sees the uncorrected data, hence Adobe also always corrects the raw data with the profile in the same way. If you want to see what your camera is actually producing then use another raw converter that doesn't have lens profiles, something like RawTherapee: RawTherapee Blog