I use a french press so it usually is sitting for about 3-4 minutes before I pour it into the mug then add milk. Maybe if I warm the milk up a little first so its not such a shock?
My wife uses plant-based milks and it seems to only happen to non-barista type milks. If it's just the plain Flax, Oat, Soy, whatever it seems to split easier than those that are specifically designed for barista use. They're not much more expensive, and she's the only one who uses it, so it's worth the extra. Your results may vary.
I’ve had it happen in the Starbucks flavoured non-dairy creamer as well. It seems to happen either when it’s been sitting a while, or if it gets shocked by hot liquid rather than a slow gradual pour (“milk” first).
I don’t recall having had it happen with Earth’s Own Barista Oat, which is what my local (Canadian) Starbucks all use, but it’s been a hot minute since I’ve had any.