It's shifted from Abolition -> Women's Vote -> Civil Rights -> Gay Marriage -> Currently in the Transgender Equality era with -> Animal Right being the next frontier.
I like your take on things but I can't help myself from observing that you have painted this with too broad a brush.
IMHO, one of the problems you might encounter is that thinking a movement culminates with an amendment or a supreme court ruling. And that the end of a movement marks the beginning of the next one. Although I think you could make a case for that if you just looked at individuals.
For instance, 150 years after the 15th amendment, we still have slavery. Not just worldwide - but right here in the USA. (1)
During the abolitionist movement, a split occurred when female abolitionists also wanted the right to vote for women. And it took another 50 years for them to get them. And its been 100 years and still no ERA. (2)
The whole 60s civil rights movement was about getting rights that were pretty much already guaranteed by laws. And any quick perusal of the headlines could lead one to believe it isn't finished yet.
And I think it is a mistake to think the Animal Rights movement is just getting started. I don't think you should even say it started in the 70s with ALF and PETA. The ASPCA goes back to the 1860s. And it other countries it goes back centuries. (3)
But if you look for parallels with the rights for women and blacks, Animal Rights are going to have a long hard road.
But on the positive side....
The animals seem to have gained some new allies. Not just silly vegans but environmentalists, progressive liberal democrats, and the labor movement.
1.
https://en.wikipedia.org/wiki/Slavery_in_the_21st_century
2.
https://www.history.com/topics/black-history/abolitionist-movement
3.
https://www.thoughtco.com/historical-timeline-of-animal-rights-movement-127594