I have a question I've wondered on and off about for some years.
Basically this, if I arrive on a flight leaves less than the stipulated MCT for my connecting flight (obviously on a single ticket) Do I really have to run myself ragged in dashing across various terminals to the connection? Or have I already "missed" the connection and thus could just go get a long cold drink and consider filing a claim for delay? I'm too old for those multi-terminal sprints that I used to do (often finding that I'd been offloaded anyway.)

boiled down to its essence. under mct equals claim compensation?

No one simple yes/no answer. Will depend on the airlines, airport and a gazillion other reasons. Cannot assume you will not get your connecting flight,

Just missing a flight due to MCT fail will not mean you get compensation. Again depends on the situation.

There is no such rule. Say your flight arrives 5-10min below MCT and your onward flight departs from a nearby gate, I bet you will struggle to get comp. for delay (because you "missed" the onward flight") when/if the agent finds out that you had to walk 2 min to the gate of the next flight (and you will be lucky if the agent agrees to rebook you to the next flight for free)

I’ve had KLM @ AMS refuse to rebook me in such circumstances and tell me to pay up for a repriced ticket or to separately buy a new ticket. I never did get around to seeing if an EC 261/2004 claim would work in such circumstances, but since the squeaky wheel is more likely to get the grease than the silent, can’t hurt to squeak and speak up.

I think that even if your inbound flight arrives within the MCT you should still go to the gate for your connecting flight (at normal walking speed).

Because it might be hard to claim compensation, if for example they were holding the flight a few minutes to allow connecting passengers to make it.

If you then arrive at the gate and boarding is closed, then it's off to the transfer desk and also file a claim online ( depending on the reason for the delay of the inbound flight).

Can anyone provide more information on this? Are there any actual published or internal rules that are used to determine if the passenger is at fault or airline when it comes to missing a connection when the first leg arrived late due to flight delay?

1) If I exit the plane down the stairs, and outside they have some agent holding a sign with my next flight, can I ignore their van transfer to the terminal and take the slow bus instead?
2) If I do use the representative am I obligated to follow his instructions to run or walk fast?
3) If I'm sitting 1A do I have to get off the plane first or can I take my time and exit last?

The reason for these questions is not to take advantage of EU 261 and purposely miss the connection flight for the 600 Euros. It's to hold airlines accountable for the delays that they could have controlled for, and for flight itineraries sold with connection times they choose to publish.

The airline is doing its absolute best to try to get you quickly on the connection flight, I commend them, however it might not be in the passengers best interests:

First reason, some people don't like to be rushed, it's stressful, and would rather miss a flight and even stay in a hotel overnight then to run through airports. Passenger books flight with 3hr connection they might have wanted to relax, shower, shop, eat, inject insulin, and while they would accept flight delay circumstances that lead to a reduced connection time equal to the MCT, they don't want a rushed 20min connection.

Second reasons, Had a 30 min connection due to delay, I've made the connection flight but my checked baggage didn't. This is much more disruptive to my travel plans as it could take days or weeks to get your baggage, and with Lufthansa you don't get any useful help. I'd rather take the next flight knowing my baggage is onboard. EU 261 doesn't help with delayed baggage.

The Montreal Convention does, though.

Your question is quite broad and there's no one-size-fits-all answer. Depending on the flight and airline, the Montreal Convention, EU 261 and the airline's contract of carriage may apply. For example, if you're flying a domestic flight in the US and your inbound leg is delayed, you're free to do 1, 2 and 3, but you shouldn't expect the airline to pay for a hotel room or for the alternate flight options to be the flight you'd necessarily prefer to take. If you think you're booking a connection that is too tight and you prefer more leisurely connections, I'd suggest booking the connection you want. I don't think most airlines will care about individual attempts to hold them accountable.