IP address characteristics for private email servers – How does an email inbox work from a "warmed" IP address to a "cold" IP address?

We send e-mail blasts to older lead lists, so we need to use private e-mail servers.

We recently purchased a new IP address, so we need to turn it on to ISPs so they can inbox better.

However, if we send an email with a cold IP to a list and send emails to that list until the IP is warm, the list will receive more inbox emails or simply follow the same inbox behavior as before The first interaction was with a cold IP?

In addition, warming up IPs tends to send emails to your cleanest and most reliable lists first. Or do you use your worse conversion lists so that you can first warm up the IP and upgrade your mailbox with the more important lists?