I'm not sure if this answers your question but I'll try.
But what happens when I am not with them?
When I was younger, my future wife asked me to think of her and phone her during the day when I was at work -- and I didn't understand, because when I was at work, then I was only thinking about work.
So that's kind of like what you're talking about, i.e. there's a proverb in English, "out of sight, out of mind".
I guess that accords with the Buddhist theory of dependent origination, i.e. that "perceptions" and "feelings" arise as a result of "contact" (see phassa).
Why would anyone care about me when I am not around them?
Nowadays I think about my mother sometimes -- "How is she, still alright? Is it time to phone again?" A Buddhist explanation for that might be "attachment" (see upādāna).
There might be other Buddhist explanations for that too, including ethics ("There are duties to mother and father"), and the Brahmaviharas more generally
(and partly a result of her karma, her being kind to me made it easier for me to reciprocate).
Buddhism does teach that "Whatever is subject to origination is all subject to cessation" -- so if "caring" for someone originates (with "contact") then perhaps it's not surprising that is ceases too.
Perhaps all I can do is care, when and while I'm in contact -- in person, on the phone, or even day-dreaming. Perhaps you might care, reliably, every time you are in contact -- i.e. "continually" even if not "continuously".
Essentially why anyone would do anything without any reason, why anyone would unconditionally care?
The question "why without any reason" doesn't quite make sense, so I think this might be more accurately phrased as three questions:
- For what reason would someone care?
- For what reason would someone care unconditionally?
- For what reason would someone care universally?
I think (though I don't know of a canonical reference) that there are several reasons to care:
- Training (parents and others teach us to)
- Self-image (we want to "be good" or "behave well")
- Reward of peace (it's more pleasant to experience harmony than emnity)
- Sympathy (I feel upset if I perceive you as being unhappy)
- Ethics
In summary I think it's to avoid some of the "suffering" that might otherwise be experienced by self and/or others, or to experience some of the "reward" for good behaviour -- including the "absence of remorse" which is said to be the basis (see What is the basis?).
As for "unconditionally" I think that means "regardless of the other's behaviour". The classic example in English is that parents' love is described as "unconditional", meaning they love their children even when the children misbehave or are annoying. A reason for that is ethics again, or strong attachment perhaps, but I think it's also perfectly rational and sensible -- if for example my wife were angry with me then it would be unwise of me to respond to that in kind, i.e. with anger instead of caring.
As for "universally" I think that means "regardless of contact or nearness". There's a Buddhist meditation practice called "Metta bhavana". "Metta" means (approximately) "caring for", and "bhavana" means (approximately) growth or development or practice. Classically (for example according to these instructions), the "caring" or "well-wishing" is something which you first apply to yourself -- then towards those near and dear to you, your teacher, your family, your friends -- then towards those who are perhaps more neutral -- then towards any so-called enemies -- and at last, universally.
