A.D. 2050 Gregorian
Temporal immigration law is difficult because enforcement is problematic. There is a whole new set of questions about "when are you from" and never mind "how long are you planning to stay?" And the border is not static, so any official encounter is going to occur in the field, impromptu and at large.
The passport is easy a brief, discreet visit with an illegal operation slightly deeper in the future and you're documented. In 2089 you can begin to get almost any known document printed at an ATM nobody cares anymore. Beware of scammers they're take your fingerprints away. Otherwise it's pretty simple.
And if you're a few decades out fashion-wise, that's no thing. We're all wearing items of clothing that we used to not be able to find anymore.
But the future is like another country, and it's important to remember that there are still going to be weirdos who don't want you there. It's not you, per se, but what you represent you're "not from around here," and you might get blamed for other peoples' unhappiness. This will not have changed much anytime.
The primary reason for moving to the future, of course, is work to get away from it.
Many up until the middle of the 21st century endured a kind of slavery, working just to live. Some were getting money just for being somewhere that they didn't want to be work for the sake of work. It was a folk belief, a vestigial behavior.
With its progressively humane and reasonable conditions including a decreasing obsession with compulsory labor the appeal of the future was obvious to some. Time-travel migration was compelling.
But there are always and everywhere going to be people who don't like immigration. It it's not "lazy people taking our jobs" (and it's not, anymore,) then it's something else. It's a human behavior, nearly universal amongst the small-minded.
For them was developed the anachronometer.
The anachronometer is a device that employs a fairly simple voice-recognition/text analysis algorithm. Its unique function is that it refers to an extensible index, a library of recorded voices dating back to the invention of time travel. Expressions and accents become definitively metric, making one's way of speaking a signature that identifies one's native chronological residence. Phrases are points on a cultural fingerprint, and any statement quickly forms a three-dimensional shibboleth.
With a basic-model handheld anachronometer, any flatfoot cop is usually able to identify a time-traveler within five or six words. It's a rare newcomer who can speak a full sentence before being escorted to the local station-house.
Of course, the future needs immigrants, too.
But ignorance has a vote and the law is the law.