Short answer: badly, just like political Christianity, but starting from further back.
I suggest that a look at what happened among political Christians in England can give us a clearer idea of the possibilities for social and political evolution in Muslim countries.
More generally, political Islam is either losing very gradually in the short and medium term to the forces of reason, justice, and human rights, or it is setting itself up to lose in the long term, either to those same forces or to violent revolution. Pretty much like political Christianity, but starting from much farther back in countries that have suffered endlessly from tyranny or empire or great-power rivalries.
To understand how that works you need to understand something of the history of Christianity in politics. The progress of political Christianity in its centuries-long contest with modern ideas of science, free speech, free thought, and so on, suggests that we should not be so impatient with the seemingly glacial social evolution of Muslim countries, and that we have to have long-term strategies for coping with the fallout in the meantime.
Udate: Renamed. I realized that my earlier title didn't convey what I meant to get at.
Now, first, we must be clear that there are many kinds of Islam (not just the larger divisions of Sunni and Shi'a), as there are many kinds of Christianity, and the political evolution of every Christian or Muslim country has been and will be different. All of the countries with neither Muslim nor Christian majorities too, of course. I am going to draw your attention to one major strand in this tapestry. Many others also reward serious consideration.
The most notable case right now is Egypt, now firmly under military rulers who have allowed themselves to be provoked into attemping the destruction of the Muslim Brotherhood. This is no more possible than the destruction of the Puritans or the Church of England or the Catholics was in England in the 16th and 17th centuries. Governments have been trying to destroy the Muslim Brotherhood for more than 70 years, ever since it first turned to sabotage and terrorism, and began to ally itself with Nazi Germany. It is stronger than ever, and just as vicious, while the rest of Egyptian society has changed radically for the better. (Hey, just like the US!)
I could easily run through the entire list of Muslim countries, pointing out both positive and negative developments. Pakistan, where the terrorists used to all work for the government, has just announced its first anti-terrorism campaign. I don't know how much progress that represents. Bangladesh, where microfinance first appeared, has its own educational computer program, comparable with One Laptop Per Child, and was the first country in the world to publish a complete set of digital textbooks online under Creative Commons licenses. The set in Bangla was published first, and has recently been followed by a partial set in English. That is unquestionably progress.
But the news of the day is not my topic today. I am proposing that we look for deeper insight. Skipping over most of political and religious history in Europe, I propose to take up the story in the time of Martin Luther's Protestant Reformation, specifically with King Henry VIII of England. This is also the time of a number of wars of religion, alternating periods of slaughter and quite grudging and uneasy toleration. This includes
- the unnamed war between German Lutherans and Catholics, ending in the Peace of Augsburg (1555)
- the Thirty Years War between all of the Catholic countries in southern Europe and all of the Protestant countries to the north, culminating in stalemate with the Peace of Westphalia (1648)
- the Eighty Years War (also coming to a close in 1648) between the Spanish Empire and the Netherlands, during which the Dutch threw off Spanish rule and economics and the Inquisition, and established Free Trade and Freedom of Conscience as founding principles of the Dutch Republic of 1581
- The English Civil Wars between Anglican Cavaliers and Protestant Roundheads, of which more below
Do you see the parallels with Sunni-Shi'a wars between Iraq and Iran, and within Syria; with Al Qaeda and Hamas and Hezbullah and the Muslim Brotherhood, and other political/religious/terrorist organizations; and with earlier liberation movements against all of the European empires? Let me give you a few more clues to work with.
The bare events were
- King Henry VIII rejects Martin Luther's advice to commit bigamy in order to get a male heir, after King Charles V of Spain, nephew of Henry's wife Catherine of Aragon, won't let the Pope give Henry an annulment to marry Anne Boleyn.
- Henry breaks with the Roman Catholic Church, creates the Church of England, and marries Anne Boleyn, whom he later has executed on multiple fabricated charges of adultery when she fails to produce a son.
- Henry has both of his daughters, Mary and Elizabeth, declared illegitimate by Act of Parliament at different times, and much later has them declared legitimate again so that both are in line of succession to the throne.
- Henry dissolves the monasteries, executes monks who resist, and distributes their property among his favorites.
- Henry dies, and his nine-year-old son Prince Edward, by third wife Jane Seymour, becomes King.
- During Edward's reign, Church of England doctrine and practice are changed from Catholic (but without the authority of the Pope) to Protestant in matters such as abolition of clerical celibacy and the Mass, and holding services in English.
- Edward dies after six years on the throne, to be succeeded by Lady Jane Grey, deposed after 13 days in favor of Princess Mary.
- "Bloody" Mary reinstates Catholic Church, burns some of Henry's Bishops at the stake.
- On Mary's death, Elizabeth becomes Queen, reinstates Church of England, and tries with very limited success to pacify Catholics and Puritans. Catholics are executed for overt terrorist plots and failed assassination attempts and for trying to convert English Protestants not only to the Catholic Church, but to attempts to overthrow Elizabeth.
- Elizabeth has her Catholic rival, Mary, Queen of Scots, executed after the Pope declares Mary to be the lawful Queen of England, on evidence that Mary was involved in plots to overthrow Elizabeth.
- When Elizabeth dies, Parliament invites Protestant King James of Scotland to be King of England and head of the Church of England.
- King James commissions the King James Bible.
- James's son Charles I loses a civil war with Parliament, in part over his assertion that he ruled by absolute Divine Right, and is executed on a charge of treason for trying to raise a foreign army against England.
- Oliver Cromwell, one of Parliament's Generals, establishes a Protestant Puritan government, eventually theocracy, with himself as Lord Protector, and dismisses Parliament.
- After Cromwell dies and is succeeded by his son, Royalist Cavaliers restore Charles I's son as Charles II. He restores the Church of England to primacy.
- When Charles II's son James II dies, two Catholic heirs are prevented from becoming King: James the Old Pretender, and Bonny Prince Charlie, the Young Pretender.
- Bonny Prince Charlie, with French backing, fails to raise an insurrection in Scotland, and flees Over the Sea to Skye (The Jacobites have all the best tunes) and then back to France. [Charlie, barley, buck and rye, What's the way the Frenchmen fly?]
- Parliament decides who the next several kings and queens will be, culminating in the Glorious Revolution, when William of Orange signs a binding contract to become King on condition that he recognize Parliament's authority over him.
- Then after William and Mary, they bring in the German Georges.
Whew!
That's without even getting into multiple wars with Scotland and France, and against Spain in the Netherlands. Then there was most of another century before the American Revolution, the Articles of Confederation, and the Constitution, followed by assorted minor rebellions and Constitutional crises, and one major rebellion (the Civil War), and so on. And still to this day a minority of bloody-minded pseudo-Christians (Protestant, Catholic, and Mormon) expects to establish a theocracy in the US. Some intend to bring on Armageddon.
So cut Egypt and other Muslim countries struggling to establish democracy and human rights some slack. The idea that you write a Constitution and hold elections and, voila, you have a functioning democratic republic is nonsense. The idea that you can explain modern Western Enlightenment-derived ideas that took centuries to establish to a population that has been suffering under tyranny and late medieval levels of religion and they all go, "Of course! What were we thinking?" is worse than nonsense. Even the idea that you can explain matters to Dominionist and Armageddonist Christians is as far beyond nonsense as trying to explain matters to the would-be martyrs of the Muslim Brotherhood.
I served in the Peace Corps in South Korea under military rule. In the decades since that time, democracy has taken hold to the point where they are not afraid to prosecute and jail a former President when necessary. Many other countries have evolved out of dictatorship, and a few out of theocracy, and it still continues.
So I have a great deal of hope for Egypt and other countries on a timescale of decades. But I do not harbor the delusion that everything will be hunky-dory and tickety-boo any time soon. How could it? For all that we have accomplished, we don't have fully functioning democracy in the US or most of Europe yet.