Its a matter of tuning in to the unique vibration of that particular Soul. Souls who have that capacity, those who have aligned themselves with doing God/Goddess intent, do not abuse that capability.
is this a capacity in all Souls,
Yes, this is a potential capacity in all Souls, to see much more of the totality of what is in front of them.
which can be increased in those who have 'aligned themselves with doing God/Goddess intent'?
I have an issue with a subtle twist you snuck in through the way you worded this, as you put it "which can be increased". You see, the wording reveals it is the personality, the ego, that is desiring to increase its personal powers. The desire is pure - a desire to serve in a more effective way. None the less, by desiring to increase a particular ability, an underlying motivation is revealed.
The difference is, a purer impulse would be to simply desire to align self ever more with doing God/Goddess intent. Then that ability, the desire to be able to see into the past of the Souls in front of it, would either manifest over time, or not. The knowing would be, if it appears it is a gift given to assist with the God/dess intended work. If it does not, it is simply not part of what is intended for the present time (or lifetime).
A parable from Ramaksrishna comes to mind. A man came to Ramakrishna and said "it's taken me 40 years but I have finally learned how to walk on water". Ramakrishna said "What a pity. You could have taken a boat for two rupees and spent the 40 years meditating on God".
Another way this is explained in the east is "phenomena, phenomena, phenomena", which means, these powers, skills, abilities are not the end goal in themselves. When looked at in those ways, they become yet another separating desire that must be released from. The idea is to focus on the root, not the manifestation.
Can such a trait be increased by diet, body-cleansing, hong-sau meditation, etc...
The point of doing purifications is to purify, not to try to attain an end result from the purification process. The whole point of purification is to release attachment to anything personal, to surrender to the higher intents. Thus desiring to achieve this or that is just one more attachment we need to drop. We each are given what is appropriate at this time to assist with the Soul work we are intended to do.
In terms of how to drop these attachments, that is the whole point of the post I wrote on how to deepen self, and how difficult it is to change. You might want to look at it again, with this in mind. The subtle attitude I am pointing out is a perfect example of a deeply existing pattern that is subtle, embedded, and difficult to change. The recent talk about total self honesty - that means facing these difficult places within us.
The goal is to align the sense of self-identification with the Soul not personality. As that is progressively accomplished, possibly over thousands of years, the rest simply happens. You can't isolate it to specific practices in a single lifetime. If you are doing these practices with an end in mind, to attain an ability like past life viewing, you have an agenda, a desired outcome. What happens if you've done the practices intently for 60 years, your life is ending, and the ability has not appeared? We have to be prepared to spend 10, 20, 30 lifetimes if necessary. Not seeking an ability, but seeking to align ourselves ever more with Soul and divine intent.
That is much closer to the way things really work.