you kill the old chain with a bunch of gradual steps. you don't kill it stone dead instantly.
you make it unattractive to mine and use by making it slow and expensive. it's a snowball effect. once a few people abandon it then more join. before you know it it's having its life support turned off.
no one thinks a move of this magnitude isn't slightly aligned with the upcoming fork?
If it's slow and expensive that it's actually attractive to mine because it means more fees for miners - and as more miners come in, the network returns to it's maximum speed of 1 block every 10 minutes. And BCH has proven that you can't easily damage Bitcoin by introducing competing chain to bribe miners with its profitability - this is hard to sustain and the market seemed to not care at all about disruptions in Bitcoin's network. Of course SegWit2x will try something similar, but eventually mining economics and the market will sort everything out and Bitcoin will be a winner again.