Well, there's the fact that it was written as an animal rights tract for adults...
Along those lines there's always Loveliness, written by Elizabeth Phelps about a hundred years ago. "It is an officer of the court! There is a search warrant for stolen property! Open in the name of the Law! Open this door in the name of the Commonwealth!" Oh god....
The animals provide the majority of the tear jerkers, but the human characters get some in too. Jerry Barker, left waiting out in the cold for hours while already sick and doing his best to care for Beauty the whole time, provides a lot of pathos, especially in the 1994 film adaptation.
Ginger's fate: After being ruined by a careless rider, her 'breath' (essentially, her trachea) is strained and she is sold down lower and lower because of her subsequent breathing difficulties, to the point that when Black Beauty sees her again, she's so broken down he doesn't immediately recognize her. All her fire and spirit that she had at the beginning is gone, and she says she wishes she would drop dead in her harness and be out of her misery. Shortly after this, Beauty sees what looks like her body.
Oh! If men were more merciful, they would shoot us before we came to such misery.
This also crosses into Fridge Horror territory, since Black Beauty never tells us the dead horse is Ginger, just that he thought it was and hoped it was. He saw a chestnut horse with a long neck and white on the face, but these are by no means rare features among horses, and it's possible that Ginger spent many more months in agony and suffering, ending in a trip to the slaughterhouse.
Even more heartbreaking are her Last Words to him: "You are the only friend I ever had."
The terrier puppies in the anecdote Sir Oliver told the other horses, about how the people cut off their soft, floppy ears. It took another hundred years, but finally ear-docking was outlawed in the United Kingdom, and is no longer being taught in veterinary schools in the United States.
Black Beauty being injured because of Ruben Smith's drunkenness, and Smith's death, leaving his wife and children alone. YMMV on how sad Smith's own death was, because he was drunk so it was partly his own fault, but it was said that Smith was generally a good man, and well-liked.
Following on from that, Black Beauty standing there with shattered kneecaps and ripped feet, listening to Smith die, praying for someone to come, and eventually simply thinking of his mother and childhood.
Beauty and Joe's reunion in the 1994 film; easily challenging the reunion scene in Homeward Bound. Bring the buckets.
Also Fridge Horror, but at one point in the story Black Beauty sees a pony that looks exactly like Merrylegs being whipped until he bled and having his mouth torn by the bit. Again, Black Beauty hopes that it is not Merrylegs, but never states this for certain.
It probably wasn't, because when Merrylegs was sold to the Vicar the conditions were that he would be cared for and never sold, and when he got too old he would be shot and buried. Of course, the Vicar may not have kept that promise...
The Fridge Horror is further fueled when you remember Beauty and Ginger were sold to the Earl of W—- because he was a friend of Squire Gordon, who hoped they would be treated well on account said friendship. Before the year was out, both were ruined (Beauty's knees and Ginger's trachea) and pawned off like dresses going out of season. While not malicious, the earl didn't make sure they were properly looked after in his home any more than he made sure they got good homes. He even laments the money he paid for them more than his disservice to a friend. Who's to say the vicar did any better? Or that something didn't happen to him (like with Jerry), and the people around him didn't honor the promise?
The last scene in the book, where although Black Beauty is safe and happy, he chooses to dream of his days in Birtwick, standing with his old friends in the orchard. The fact that he never even saw all but one of those friends again, much less say goodbye to them, is somewhat painful when you consider it is very likely that Black Beauty is the only one still alive and the world he lives in is generally not kind to horses.