Should brake lever move if you press the (V-brake) calipers together? The &quot;normal&quot; way is pretty clear: you press the brake lever, it pulls the bowden, which then forces the V-brake calipers together.
However, the question is: should it work also the other way round? I.e. should manually pressing the calipers together also cause the bowden to slide inside the housing, and hence pull the lever on the handle bar?
Case in point: yesterday I replaced my front brake calipers and bowden/housing on my bike, and observed the mentioned behaviour (which I am 99% sure was not the case previously). The brake lever does not go all the way, but it does move considerably (I would say 20-30% of all the way).
I can interpret this in two ways:

I did something wrong during the process (maybe not fixed well enough something, which should have been fixed?). In particular, I am concerned about the lever being too lose (I intended to replace it as well, but unfortunately, the replacement I bought turned out to be too small.)
Since the bowden/housing is new and greased, maybe the bowden can just slide more easily, so the force, instead of bending the housing (which was probably the case before), makes the bowden slide smoothly, up till the lever.

Some more thoughts:

On the old rear brake caliper, this does not work. The bowden for the rear brake has 3 sections: the first one, which runs from the lever to the top tube of the bike is inside a housing. Then, it runs &quot;naked&quot; (without any housing) below the tube, and the last section (which connects to the calipers) also has a housing. If I press the rear calipers together, then the wire slides up to the &quot;naked&quot; section, but then it bends at the &quot;covered&quot; section of the front.
In case the answer to the original question is &quot;yes&quot; (i.e.: yes, with a well-greased bowden,  pushing the calipers together should move the lever), can this be used as a test to tell if the bowden needs greasing (&quot;press the brake calipers together: if the lever does not move, you need to grease the bowden&quot;)? And if so, would this test work also for the rear brake, or only the front brake? (In case of the rear brake, I can imagine that due to longer bowden, and the covered/naked sections, the force will always cause the cable to twist at some point before moving the lever, no matter how smooth the bowden runs inside of the housing.)
I think you have the right idea. When you push the brake arms together, you are shortening the system, without changing the inner wire length. This in effect makes the inner wire 'longer', and because it is fixed at one end, it wants to move up towards the lever, exerting a force and causing the lever to move.
It seems likely that the low friction of the new housing/cable is helping to send all of the force to the lever end. I suspect that if there was more friction somewhere, you would see a different result, maybe the noodle would pop off the end of the housing, or the housing would come out of the lever at the barrel adjuster, but the housing would find some way to accommodate the relative increase in length of the inner cable, like this perhaps. At the moment, the lever itself is giving the least resistance and pivoting, perhaps this will always be the case.
The effect is further evidenced by you observation with the rear brake. Because the housing is interrupted and there is bare cable exposed, it is able to flex and bend into a curve so that the 'increased' length is accommodated and the cable slackens. This can actually be a useful aid if you want to disconnect a brake or derailleur temporarily, without undoing the cable pinch bolt. A kind of quick release.
On the front brake with uninterrupted housing, the cable is constrained by the housing and is unable to flex, so acts more like a rod, pushing the lever.
All of this, I think, is just to say 'yes' to your own observations and explanation, although I would be doubtful of its diagnostic merits, its just a fun thing you can note. Visual condition of the cable is more useful as well as the lever feel and responsiveness as you use the brake. these will degrade a little as the cable deteriorates.