Okay.
I'm a 'new partner'. I've been married to my husband for nearly three years, we've been together for four-and-a-half years, I've known my stepdaughter for over half her life now, and I've had a total of three conversations with the ex-wife.
I'm still 'a horrible person', I still tell my husband 'what to do', I'm still the cause of all her problems, I'm still 'not to be involved in anything to do with' my stepdaughter. I also lead a secret life of violence, drug use and child abuse. You name it, I've done it (according to the ex-wife).
Mate, that road never ends - you will never be able to stop your ex from rubbishing your partner.
But you can control how you respond.
And in my experience, the best way to respond is to laugh among yourselves, brush it off, and then get on with the business of parenting.
Don't respond with anger, or gallant defensiveness, or logic or reason. Getting a rise out of you is what mum wants, so don't fuel her fire.
Instead, sit on it for two or three days, then decide if you actually need to respond at all. My husband ignores nine out of every ten text messages from his ex-wife, because many of them are just criticism thinly veiled as a parenting question, and the rest are just outright emotional drivel of no relevance to actual parenting.
If you decide that you do need to respond, then put on your most pleasant pants possible, and practice assertive communication.
'Hi, thanks for letting me know X was sick. I will speak to X's teacher about possible bullying, and it may help X to speak to the school counsellor, as well. Let me know if you have any other thoughts on this. Thanks.'
Don't even acknowledge the deliberate attempt to bait you by addressing it to your partner.
Most parents who exercise kindness, even in the face of hatred, generally find that little by little, the other parent will start to mimic their behaviours.