[Website migration] Client requested i carry out seo (content & tech) during a website migration with old & new sites being live at the same time. Im kind of out of my depth here. Any advice?

The client is conducting the migration not at once but in spurts meaning both old & new websites will be live.

Client reasoning here is that gradually fading in elements of the old site into the new site will allow us to quickly accrue seo benefits since it can take some time.

Now im not too familiar with how migrations are carried out per say so i cant connect it with seo considerations at the moment.

I've always just worked on websites that were done with migrations before conducting an audit. Its the way its always been done by me and doing it during a migration introduces what i would imagine to be unbelievable complexity in both tech & coordination that may result in catastrophie seo wise

Id appreciate any advice and resources

Thanks!