Hello,

This command works to test whether a player is in the minecart or not :

/testfor @p {RootVehicle:{Entity:{id:"minecraft:minecart"}}}

This command works to advance the minecart of 5 blocks, with or without the player in :
(see the picture number 1)

/tp @e[type=Minecart] ~-5 ~ ~

This command works to change the rail minecart, with or without the player in :
(see the picture number 1)

/tp @e[type=Minecart] ~ ~ ~-2

When i teleport the minecart alone over a long distance, it works.
The minecart is well placed on the rails more than 100 blocks from where it was :
(see the picture number 2)

/teleport @e[type=Minecart] -323 11 419

When i teleport the minecart with me in, on a long distance, it doesn't work :
(see the picture number 3)

/teleport @e[type=Minecart] -323 11 419

Since i can't move and the minecart doesn't answer to the keys. So, i decide to delete it !.
Just after executing the command, i appear at my starting point but with the difference that the block below me is shifted of one block to the right !?.
(see the picture number 4)

/kill @e[type=Minecart]