Authoritarian regimes depend on a small privileged class to maintain order
In order to control a population, authoritarian leaders only need a small core of loyal collaborators. They keep those collaborators in line by providing them with a higher standard of living than they could possibly have through their own effort.
This is done by rationing goods, and then either redistributing them to friends as gifts or threating to confiscate the wealth of enemies. Since rationing cannot be sustained in a free market, because it is possible to find sellers and buyers without going through an intermediary, authoritarian regimes cannot exist when markets are free.
Now, people generally do not like authoritarian regimes to begin with. But when they make decisions that annoy great powers, the world gets together and decides on regime change through military means or economic sanctions.
Historically, sanctions have been ineffective at toppling regimes
Economic sanctions are trade barriers and restrictions on financial transactions applied by one country (or group of countries) on another country.
The rational is to make the population suffer so much, through shortages of essential goods, that they will demand reforms and even regime change. Targeted measures, such as the seizure of foreign assets, can also be imposed on those inside the regime, to force the leader to resign.
Despite their widespread use, no statistical study has shown that sanctions achieve their desired policy goal.
Intuitively, the link between international cooperation and sanctions success seems obvious. Empirically, however, the results are rather surprising. Repeated statistical tests show either no link or a negative link between cooperation and sanctions success.
Popular failed implementations include Saddam Hussein in Iraq, Kadhafi in Libya, Hugo Chavez in Venezuela, and of course the 40 year sanctions against Cuba's Fidel Castro. Still, they remain popular to western politicians because they show that they are doing something without committing to expensive military action.
Sanctions increase the financial risk of opposing the leader
Sanctions instead of weakening the leader, change the economic structure of the target country to one where the leader is able to control the redistribution of wealth and goods. This increased control of economic activity is partly the reason why leaders of country under sanctions accumulate enormous wealth (i.e. Saddam Hussein, Mobutu Sese Seko, Bashar Al-Assad...).
Whereas before there were too many merchants to control the flow of goods, afterwards, it is much easier to track and demand a cut on the sell of goods imported by the handful of contraband dealers willing to risk defying the sanctions. Leaders can then consolidate their power by giving them to collaborators who best tame revolts.
Even targeted sanctions increase the leader's stronghold. Before collaborators could maintain a semblance of financial independence by having international businesses, after the seizure of their foreign assets, they rely entirely on the leader.
Free trade reduces the financial risk of opposing the leader
The solution to toppling a regime is therefore not to restrict trade, which concentrates redistribution powers in the hands of the authoritarian, but to facilitate it.
By encouraging trade, collaborators will be able to have a higher standard of living through their own effort than the one the leader can provide. They will as result work on their own projects while the regime, due to a shortage of loyal officer to keep order, will decay by itself.
There is precedence. The most impressive is the Soviet Union, which shot itself in the foot in May 1988 when it implemented the Law on Cooperatives. It allowed, for the first time since the communist revolution of 1918, the private ownership of businesses in the services, manufacturing and foreign-trade sectors.
This eliminated the state monopoly on production and redistribution. Consequently, the state completely lost its hold on its people who regained a sense of independence and entrepreneurship. Three years later, in 1991, the unshakable Soviet Union fell.
The South African case
The only example of sanctions leading to the overthrow of a regime of a mid-sized country was Apartheid in South Africa.
In the mid 1980's, the developed world imposed gradually stricter economic sanctions on the racist South African regime. It was to punish the regime for denying human and voting rights to black citizens who made up 80% of the country's population.
The economic sanctions were weak as exports rose 26% from 1985 to 1989, and their impact were estimated at only 0.5% of GDP. The sports obsessed South African also saw their national teams be also banned from international competitions. However, they had a strong psychological impact on the country and in 1991 the Apartheid regime was replaced by the Africa National Congress.
Sanctions worked in South Africa, partly because as a democracy the regime could not redistribute resources to a small network of collaborators tasked with keeping the peace through extra-judicial means. The Apartheid regime had to bow to the demands of white voters who were ready to grant to all citizens basic human rights for a return to economic normalcy.
However, today's sanctions are imposed on authoritarian regimes that depend on a small core of loyal collaborators to maintain order. The leaders of Cuba, Syria, North Korea, Iran or Russia only stand to gain from sanctions because their henchmen will be forced to depend even more on them.
Relaxing sanctions, or even helping local merchants trade internationally, would greatly diminish their leaders' stronghold on the economy and society.