Unfortunately, how Gil time travels in the movie is not disclosed and Woody Allen never explains the mechanics of his time travel as any explanations in the script would take away from the plot of the movie. We are to just believe that something magical happens around midnight.
I have seen this movie several times and the following is how I interpreted the time travel aspect:
Gil possibly unknowingly came upon a wormhole at or around that particular street at or around midnight. The arrival of the car is just the transport mechanism. Remember that Gil returns to present day after speaking with Hemingway and leaving the cafe to go and retrieve a copy of his book. He was on foot, not in the car. He walked away for a moment only to turn around to see that the cafe from the 1920's had turned into a laundromat in present day.
I believe that Gil is only transported back to the 1920's from this particular wormhole due to his intense desire and almost obsession to actually live in that particular time period coupled with the fact that he is obviously miserable in present day with his job and his horrible fiancee.
When Gil meets Picasso's mistress, Adriana, she expresses her desire for the past, particularly Paris in the 1890's. Remember that both Adriana and Gil both inadvertently travel to 1890's Paris where Adriana chooses to stay.
I believe that the wormhole lets the traveler represent their innermost desire about the particular time period that they are being transported to as well as reveal character strengths and weaknesses.
As for Tisserant, there could have been tangled strings, if you will within the wormhole that did not allow him to follow or see Gil in the 1920's. Also, Tisserant had no great passion or desire for the 1920's. Perhaps he had a hidden passion for 18th century Versailles? (I am merely speculating the exact time period and location). I believe that Tisserant is shown as being left in this time period as more of a comic relief than anything.