Depends on what part of the French Revolution we're talking about! This is gonna be long, so will break it into two posts:
The French Revolution had periods of changing leadership that trended towards more liberal (radical) until the Thermidorian Reaction, aka fall of Robespierre, in 1794, when this trend reversed until Napoleon and the coup of Brumaire, which is a popular end date for the French Revolution for historians.
One other thing to note before we dive into your question, but the French Revolution was characterized by the desire of the revolutionaries to implement liberal, Enlightenment philosophy, but hitting the realities of governance. There were many times that they had to choose between implementing their principles blindly in a vacuum, or compromise these philosophies by taking into account the current climate. This explains, to me, a lot of what people commonly see as the 'hypocrisies' of the French Revolution. A great example of this is the heart of your question: the limitation on peoples movement into and out of France.
So, to get to your question, let's start with the ancien regime, aka pre-1789 France. To quote Peter McPhee in Liberty or Death: "Under the ancien regime foreigners could live freely in France, albeit with restrictions on whether their property could be passed on." In fact at the beginning of the Revolution we can see many such foreigners living in Paris. The port cities of France also were bustling, due to France's involvement in the slave trade and their cultivating of the crown jewel of their colonies, Saint Domingue i.e. modern day Hati.
Moving into the first phase of the revolution 1789-1791, the political atmosphere was dominated by what we can call the "Liberal Nobles". These were men like Lafayette and Mirabeau who were formerly nobles under the ancien regime, but were 'Enlightened' on questions such as equality before the law, the natural rights of man, and importantly religion, as their more liberal attitude towards the Catholic Church in France separated them from more conservative liberal nobles, and certainly alienated the King, and gave their 'faction' a very tenuous hold on power. This was the period of the French Revolution with the most pure, unadulterated application of political Enlightenment philosophy (note: they did not attempt to answer any social questions, like the right to exist or the right to work). The King was still nominally invested in the Revolution (or so it seemed), France was not yet embroiled in war, and the atmosphere was generally one of joy at what most saw as beneficial changes to a kingdom that had been dogged by stagnation and failed attempts at reform for years.
During this time, there was a more-or-less 'open door policy' in and out of France, because a literal liberal interpretation of the rights of man means you cannot control where a person choses to go- that would be infringing upon their liberties. This was characterized by a mass exodus of nobles (mainly Courtiers) after the fall of the Bastille in July of 1789, lead by the King's brother the Comte D'Artois who saw that Louis wasn't going to crack down on the Revolution and said "nah yeah I'm done".
The first event that precipitated a debate on the freedom of movement in the National Assembly was the King giving his elderly aunts permission to go to Rome in February 1791 to celebrate Holy Week and to confer with the Pope about the Civil Constitution of the Clergy. This precipitated a debate in the National Assembly about the freedom of movement, with some, like Mirabeau, arguing that their liberal principles prevented them from not letting two super old ladies go to Rome. Others saw the situation a bit more practically, and worried that this was just the start of the royal family getting the f out of France so they could come back at the head of an army of foreigners and emigres (remember, Marie-Antionette's brother was the Emperor of Austria) to crush the Revolution. Louis personally saw to their travel plans, and the crowds were too late to stop them from going.
The second event that started to show the revolutionaries (at this point assembled as the National Assembly) that freedom of movement might not work in practice was the Parisian crowd stopping Louis & his family from leaving Paris to celebrate Easter at their chateau in Saint-Cloud in April 1791. Now if you'll recall Louis & the gang weren't living in Paris because they dearly missed being so close to their people- they were more-or-less forced to do so by crowds of women, later backed up by the National Guard, during the October Days in (duh) October 1789. Coming hot on the heels of a few iffy events in the eyes of the revolutionary mobs (the infamous Day of Daggers, the fact that Louis was still taking communion from a 'refractory' priest, i.e. a priest who had refused to take the Civil Constitution of the Clergy), the Royal families attempted 'flight' to Saint-Cloud was not gonna happen. As soon as they were in their carriage a mass of people surrounded it, preventing them from leaving. Lafayette showed up and attempted to disperse the crowd, but his National Guardsmen refused the order. For two hours the Royal family sat in the carriage waiting for the crowd to get bored I guess, or for the guardsmen to have a sudden change of heart, but as neither happened they went back inside, proving to Louis once and for all that he was in fact a prisoner in the capital.
Okay so now the biggie, the King's infamous Flight to Varennes, or as Timothy Tackett says it perhaps should be called, the flight "as far as Varennes", because the king was trying to get to the army garrison in Montmedy. This was in June of 1791, and Louis most unwisely left behind basically a 'Top 10 Reasons the Revolution Sucks' list, so when the gov't tried to spin it as a kidnapping that was pretty damning evidence to the contrary.
Passport systems had already been nominally in place starting in 1791 (for example Louis gave his Aunts passports to go and visit Rome) but things really heated up after the declaration of war on Austria (which Prussia joined shortly thereafter) in April 1792. At this point the 'universality of man' was giving way to (not entirely unfounded) suspicions about conspiracy and foreigners plotting in their midst. One important thing to understand about the French Revolution is the widespread belief amongst the Revolutionaries in the philosophy of Rousseau, and the innate goodness of the people (le peuple in French, which translates to a bit more of a collective feel, as in le peuple of the patrie i.e. fatherland). If the Revolutionaries were stressing the goodness and virtue of the French people, saying they had just been corrupted by the nobles and priests, they can't really turn around and blame insurrection and counter-revolutionary speech on those same peoples. So the common thought was that foreign agents within France, along with former nobles and non-juring (aka refractory) priests, were stirring up trouble and corrupting the goodness of the people. Now I usually see these conspiracy theories posited in the light of "psh silly revolutionaries so paranoid" but it's important to note that several attempts were uncovered, usually instigated by Louis' emigrated brothers the Comte Provenance (who had successfully escaped when Louis & fam had been caught in Varennes) and the Comte D'Artois (who as you recall had gotten out after like the first Revolutionary event).