Although the movies are set almost 70 years earlier, a fair number of characters from the original Harry Potter series were alive during the events of the Fantastic Beasts: The Secrets of Dumbledore takes place, but it's likely sometime during the 1930s.
Given that five total Fantastic Beasts movies are planned, some time jumps will be necessary in order to reach the duel between Albus Dumbledore and Gellert Grindelwald before the franchise's conclusion. This infamous duel, which helped cement Albus' lasting fame and ended a Dark Wizard uprising, occurred in the year 1945. By that time, even though many of them likely won't make an appearance, several beloved characters from the original Harry Potter series will have been born.
The Harry Potter movies are set in the 1990s and end with the 1998 Battle of Hogwarts during Harry Potter character who was alive during the Fantastic Beasts timeline.
Albus Dumbledore
Albus Dumbledore is only one of a few Harry Potter characters who actually appear in Fantastic Beasts, since his past is a prominent plot point in the series, which has been slowly setting up for his duel with Grindelwald. Although Albus' past accomplishments awarded him great fame and respect from most throughout Harry Potter, he is often treated with suspicion and even disdain from his superiors in Fantastic Beasts. Hopefully, the movies will fulfill their promise to show the wizard at his prime, performing the kinds of magic rarely seen in Harry Potter. In the past, Rowling has stated that Albus Dumbledore was about 150-years-old at his death, but later claimed he was born in 1881, which would make Albus closer to 115-years-old at the time. In Fantastic Beasts 2, he is about 46-years-old, and will be 64 for his infamous duel at the end of the prequel series.
Aberforth Dumbledore
Since Aberforth Dumbledore is only a few years younger than Albus, he would also be alive during the Fantastic Beasts' timeline. Considering how much effort Albus put in to hide his past and his family from public scrutiny, it's unlikely that his brother would make an appearance in Fantastic Beasts. They did not get along very well for much of their lives. Despite this, Aberforth helped students and Order of the Phoenix sneak in and out of Hogwarts during the final Harry Potter film, proving that despite the bitterness he felt toward his late brother, it did not prevent him from doing what he could to stop Dark Wizards.
Voldemort
Voldemort was born on December 31, 1926, meaning he would be alive prior to Fantastic Beasts 2. Of course, the boy known as Tom Riddle would not become very relevant to Wizarding history until after his graduation from Hogwarts, at which point he would be free to pursue potential Horcruxes full-time, as well as start seriously growing his number of loyal followers. Despite his power and influence, eventually becoming the most dangerous Dark Wizard since Grindelwald, Voldemort would do his best to avoid Albus Dumbledore as much as possible.
Rubeus Hagrid
Rubeus Hagrid's birth in 1928 means he would be alive by Fantastic Beasts 3. It's possible that Hagrid could make a cameo appearance in a later Fantastic Beasts film, since he will have started at Hogwarts a few years before the final movie. As a lover of all magical creatures, Hagrid would be an ideal Harry Potter character to show up in the franchise. He would gladly help Newt Scamander care for a creature in need — in fact, such an encounter with Newt may even be the spark that begins his lifelong appreciation for all magical beings.
Minerva McGonagall
Sources differ on Minerva McGonagall's official age. In The Harry Potter and the Order of the Phoenix book, McGonagall revealed that she had been teaching at Hogwarts for 39 years. Since J.K. Rowling had confirmed she began teaching at 21-years-old, and The Order of the Phoenix is set in 1995, that would make her birth year 1935. Yet, a young Minerva McGonagall appears in Fantastic Beasts 2 to be already teaching at Hogwarts, making her many years older than what had previously been considered canon. Whether this was simply a plot hole, or has some sort of explanation that will be cleared up later, Minerva is definitely alive by the end of the Fantastic Beasts series. Just how old she would be at that time, however, cannot be known for certain.
Bathilda Bagshot
Like many Harry Potter characters, Bathilda Bagshot's exact age is unknown. When the Dumbledore family first moved to Godric's Hollow in 1890, Bathilda was already an adult, making her at least 17-years-old at that time. During the Fantastic Beasts movies, she would likely be in her fifties or older. Bathilda is best known for writing A History of Magic, which is taught in Hogwarts' magical history class. She is one of the oldest, if not the oldest character in the original series.
Elphias Doge
Elphias Doge, born sometime between 1880 and 1881, was a close friend of Albus Dumbledore, and responsible for writing his Daily Prophet obituary when he ed. He and Albus went to Hogwarts together and were quite close, both considering themselves outsiders at the school. They had planned to travel the world together after graduation, but the sudden death of Albus' mother derailed those plans, so Elphias traveled alone. His adventures meant he was not present for Albus Dumbledore's duel, and will likely be absent for much of the rest of Fantastic Beasts, but he did later become a member of the Order of the Phoenix with him. Aberforth Dumbledore spoke rather critically of Elphias to Harry, saying he "thought the sun shone out of [his] brother's every orifice." Like many wizards, Elphias Doge saw only the best in Albus, either unaware or dismissive of his flaws.
Horace Slughorn
Horace Slughorn taught at Hogwarts with Albus Dumbledore and was one of Tom Riddle's teachers. The Potions professor repeatedly delighted at Tom's clear talent and potential, and Slughorn's knowledge of Horcruxes led the future Dark Lord on the path to pursuing them. Horace Slughorn's exact age is unknown, but he was at least 15-20 years older than Tom, meaning he was certainly alive during the Fantastic Beasts franchise. Horace may even appear in a future Fantastic Beasts movie teaching alongside Albus and Minerva at Hogwarts, or socializing with those he hoped would eventually make a name of themselves.
Mykew Gregorovitch
Mykew Gregorovitch is another Harry Potter character alive during Fantastic Beasts' timeline, as he was a known wandmaker by 1926. He was not very present in Harry Potter, only really becoming relevant during Voldemort's search for the Elder Wand. Gregorovitch was rumored to have possessed the wand at some point, and was tortured by Voldemort for information on its location. Mykew Gregorovitch and Garrick Ollivander were the two most celebrated wandmakers in the Harry Potter series.
Garrick Ollivander
Garrick Ollivander was renowned in Harry Potter as being quite possibly the most skilled wandmaker to ever live. Although Harry always found the man's eccentricities a bit off-putting, he was praised for his knowledge for a reason. Ollivander was born around the year 1907, and would be a young man during the first two Fantastic Beasts movies. Since Ollivander was the one to suggest to Voldemort that Gregorovitch possessed the Elder Wand, it's probable that he never knew about the wand's new ownership with Albus Dumbledore. This makes it unlikely that he will appear in any Fantastic Beasts and Where to Find Them film, since he was not aware of the Elder Wand changing hands first to Grindelwald, and then to Albus. Considering his level of knowledge, he would have probably recognized the wand on sight.
