When Giovanni steals a time capsule and modifies it to take him back to a specific point in time, stealing a small portable storage system device called a Pass(that may or may not be used to travel to the MLP universe), it is up to Blue to find him and stop him from trying to capture the ponies and sell them for profit.
And when there's a chance to capture a legendary, it's win win. Plus, Blue knows if he didn't join Red to travel back in time, he would miss out on the chance to get his own rare pony Pokemon. (That Red managed to get one of but he didn't)
That...And Giovanni has stolen all the badges that belonged to Blue's gym.
Again...
Blue is planning on getting a Pokemon-proof safe once this is all over.
Red gets transported to Equestria by a failed experiment by professor oak.
Three Pokemon trainers crash land in the palace of Celestia, a unicorn in tow.
Lyra has been waiting four years for his return. She wasn't expecting him to be dumped at her feet like a sack of potatoes.