First off, thinking of the whole phenomenon in terms of monarchy is a little misleading. We tend to think of monarchies as hereditary, but they really weren't in this period. Tyrant is generally the word used as a close transliteration of the Greek term, but again, that comes with a lot of baggage.

So at the beginning of the archaic period, oligarchies are pretty much the order of the day. You have a bunch of small self-sustaining cities starting to shift their economies towards more international trade. Some families have a little extra which they can then invest in riskier ventures (like trade), if it pays off then they can make a good profit so you end up with a number of families with more resources than the rest.

All this is to say the oligarchs come FIRST, then the tyrants, and the tyrants never quite manage to completely disable the existing system. Around 670 BC or so, we start to see tyrants emerge. Usually, they're from a wealthy background, though often not the very top of the scale or marginalized in some way. At this point in time, the aristocratic families were continually vying for supremacy, often in ways that were very damaging for the poor. They would use their personal wealth and popularity to gather up armed crews which they would then use to take power and keep the aristocrats in line. In an interesting twist from our modern concept of tyranny, early ancient tyrants often catered to the poor at the expense of the rich, establish infrastructure and paying for food so that the masses would continue to support them.

But that isn't the part you asked about. How did they fall? Well, personal wealth and charisma got them the support of the people. Their heirs... didn't do so well. They didn't have the achievements that got their fathers in power and by the second and third generation were more reliant on their family name than their own merit which gave other aristocrats plenty of room to stir up coups and resistance. Pretty much no family of tyrants made it past the third generation.

But it still changed things. The poor now expected their governments to provide certain necessities, they expected accountability, and they knew that if things got bad enough, even if you couldn't afford a set of armor, you could still cause enough damage to change things.

And a quick post-script on the Dorian invasion - probably never happened, if it did, then it certainly wasn't as important as the later Greeks believed.