Overall:
This is a pretty good concept, and I like the premise, but I feel there are some inconsistencies and things that don't quite make sense. I think the narrative of having alternate versions of the travelers, who were once agents at the Foundation in different realities, be shadow beings that exist on a psychological plane and have the ability to drag the travelers between dimensions feels pretty disjointed. The psychological plane aspect doesn't quite gel with the interdimensional aspect. It's also mentioned that these shadows all shared the same fate, being fazed out of reality, but that's also too loosely connected to the psychological plane.
This is all without mentioning that the involuntary hopping between dimension concept is pretty specific and had kind of already been done by SCP-507. While the shadow aspect would be new, I'm not sure it's enough to set this apart.
Specific:
The entities are meant to be alternative versions of them who served in the SCP at different levels, but all had the same fate, they got phased out of reality.
I'm not sure the entities have to be alternate versions who worked at the Foundation. I'll cover this a bit more later, but I think it would clear up a lot of the more loosely connected concepts if the psychological realm was just another plane of reality.
I was thinking that the prosses by which they shift could relate to separating the atoms of their bodies, allowing them to phase between solid objects. However, the denser the object the harder it is to phase from it, allowing them to be contained in lead or other dense metals.
This feels like it's getting a little too spread out. Adding in too many ideas or properties can harm a concept, so I'd cut the phasing and focus on the shadow aspect. Also, as someone with a decent understanding of particle physics, I can say that simply spreading out or quickly vibrating atoms doesn't directly cause phasing and definitely doesn't relate to hopping dimensions.
Advice:
As I mentioned earlier, I think a lot of the problems with this concept can be solved if the psychological plane wasn't related to the mind, but is instead another plane of reality inhabited by shadows. In fact, due to the similarity this has to SCP-507, I'd actually recommend shifting the focus away from the dimension hoppers and onto the shadows.
Following this narrative (the shadow realm would be the main SCP), perhaps the shadows aren't former Foundation agents, but are instead just the denizens of this other world. (Maybe the shadow world was created by the Foundation's or another GoI's experiments) Like you laid out, the shadows want to cross over into our reality and inhabit our bodies, but instead of simply pulling us into other random dimensions, perhaps they pull us into the shadow world.
The narrative could focus on the international group that you planned, as the Foundation has gathered up people who are being targeted by their shadow, as well as how the shadow realm came to be and why the shadows want to inhabit bodies.
The main hook would be the mystery surrounding the shadow world's origins, as there may be hints that it was man-made. The article could conclude with the Foundation finding the answer to this question while trying to protect those that are being targeted by the shadows.
I know this is a rather different take than your original, but it's just my suggestion, as I tried to incorporate the best elements of your idea while shying away from the dimension hopping.
