I'm going to be following my own extrapolation of the research... but I think it's valid.
Researchers have shown in the past that exerting self-control on ourselves is exhausting (wipe-out number 1 for parents).
Researchers have shown in the past that exerting self-control in one situation makes it harder to hold it together in another situation (even unrelated - hence the difficulty-to-impossibility of holding it together ALL DAY LONG).
Researchers have shown that it's exhausting just picturing ourselves exerting self-control (even trying to gear ourselves up and prep for the next scenario is a wipe-out).
AND, now, researchers have shown that it's just as tiring to picture someone else exerting self-control (cough-child-of-mine-cough).
Interestingly, the loss of self-control was tested against an over-indulgence, and the exhaustion is on memory and skill (like word game challenges). So (by my extrapolation) the more we picture ourselves in our child's shoes, the harder it is to hold a tough disciplinary stance - we just don't have the energy.
Now, the researchers consider this a potentially bad thing. And it definitely does have its risks. However, they also didn't test for whether the loss of self-control is anti-empathetic or empathetic, because their challenge only registered a more-empathetic loss of control (paying more for a car).
Actually, the challenge they presented really was one of close empathy. It was getting into the other person's shoes and experiencing the problem from the other person's side. That engages a different part of the brain than the intellect/word-problem solving side. I wonder if the 'tired brain' thing is more 'working emphasis on the wrong area to handle logic' than 'can't think!' Definitely a bias in the study toward 'exhausted = boundaries change due to empathy/kindness overload'.
Certainly, when I put myself in my child's shoes, I am more likely to give them some slack on various situations that arise later. It feels like I'm doing it because I'm more in tune to what they want, but the research indicates it may be more that I'm less prone to holding a hard line for anyone (myself included) if I've expended energy being empathetic about a self-control situation.
It becomes an interesting question - why did I really blur the boundary of that rule? Did I really let Miss M have more Easter candy because I'd read the article by the dentist saying a single blowout day on candy was less risky for their teeth than a little excess every day for a month? Or did I let her have more because I'd spent the morning dropping myself mentally into her (and Mr B's and Miss R's) shoes as she (they) resisted the forbidden items that other kids present could eat? Was I wiped out on self-control and just had no energy to fight that particular battle anymore? Is this process where the idea of 'picking your battles' comes from?
Certainly I had no heart for the battle, at that point. So maybe the tired-out-by-exerting-and-imagining-self-control played a role, and I just found a reason that seemed valid to excuse the lack of desire to hold that line hard. I know that everyone is prone to that particular process - we are already resisting doing something, but don't know why, and then our mind supplies a reason - but the 'reason' may not have been the cause, just something to hang the resistance on. It is often valid, but it also is often just the most logical excuse, too. There's usually something else down there in the murk that is a deeper reason. On the negative side, it can be simply 'I don't really feel like dealing with this right now.' And there we are, right back at the research again.
Either way, that 'Oh, fine, whatever, I give up' feeling may be partly due to spending a lot of time empathizing - or even just picturing our kids exerting some self-control (hope springs eternal?). Not that I think we should stop doing this - seeing it from their side allows me to figure out different answers (often with the more social/emotional side of my brain, which wasn't tested in this study). But maybe knowing that I may be more prone to putting extra slack in the line after having done so could help me determine more accurately where I really should give some slack, and where I'm just prone to doing it because I can't muster the strength to hold the line any longer.
And this may also explain why the peak-points in the day (morning and evening) are so hard to maintain self-control on. It's a constant cycle of actually maintaining self-control alternating with working the kids through some self-control (with coaching, which means trying to see it from their side). No wonder it's a wipe-out!