I've never had animals stray when I was close, it's so easy to get way ahead of them though -.- They get stuck on everything and even normal walking speed is way too fast if they don't get stuck on anything.
That said, in my last vanilla survival game I managed to get a mob of every type + several villagers to my survival island (hundreds of blocks from the nearest land) without using portals (you couldn't transport anything through portals when I did that). The villagers were the only hard ones to get to my island (I had to dig a long tunnel and transport them in minecarts, destroying and relaying track over thousands of blocks), the animals were easy, they just took a long time (nowhere near what the villagers took though).