Air carrier Captain reported descending below a minimum altitude during arrival to DEN after ATC issued a last minute change of landing runways.

Narrative

We were on the TBARR 3 Arrival to Runway 7 in DEN. Keep in mind we were all set up for the TBARR Runway 7 transition and approach. Around the TBARR intersection; I heard DEN Approach changing the runways to a south flow for those in front of us on the arrival; and as always; they wait until the last second and expect everyone to be able to comply. VERY FRUSTRATING; and it happens all the time here in DEN. We had not been given the new runway; so I began asking if we would be getting [Runway] 16L as well; since now we were approaching MNARK; which is the branch point for going to Runway 7 or going to Runway 16L. It was at this time; after asking; that they confirmed the arrival and runway change. We were now almost on top of MNARK. We start scrambling to reprogram the FMC along with re-brief the arrival and approach; since we are now supposed to go to EOLUS rather than SUMTT. Because of the scramble to reprogram the FMC and brief; the VNAV disconnected. The aircraft continued to descend to somewhere around 11;000 ft. to 12;000 ft.; at which time; I told the FO (First Officer); pilot flying. We descended too far and climb back up to 15;000 ft.; which is the minimum at EOLUS. I then told the Approach Controller of the error and that we were climbing back to 15;000 ft.; at which time; he indicated something to the effect of; 'That's OK; it has been that kind of night and we can stay at 12;000 ft.; then comply with the restrictions at CLFFF for the RNAV to 16L.' Simply put; we were low on the arrival and missed the EOLUS crossing restriction of 15;000 ft. to 17;000 ft.What is frustrating is the fact that this never should have happened in the first place. I understand; had the change happened out by BBRRO or LIFTE; but NOT essentially at the branch point MNARK. The winds were calm and DEN should have never changed our runway at that point. Just let us continue and land on Runway 7 and start changing the arrivals and runways for aircraft way back on the arrival. I would understand if the winds had shifted dramatically but the winds were essentially calm. Let us continue to Runway 7 and have aircraft 30; 40; 50 NM behind us change arrivals and runways. But changing our entire arrival and approach at the last second or when you are a couple of miles from a branch point is setting us up to fail no matter what. And this kind of thing happens ALL THE TIME here in DEN for some reason.