Post transfer of shares from demat to demat

This is about what to do after the transfer of shares from one demat to another. I'm thinking of moving from Groww to Zerodha due to the increased brokerage on Groww. I searched and found that the average price of the stocks and purchase history doesn't carry over. Now I wanted to ask if brokers give you the option to do so. I found [this](https://support.zerodha.com/category/console/portfolio/holdings/articles/how-to-update-buy-average) for Zerodha, and even with this, can we enter multiple transactions for the same stock (purchase on different dates). If anyone has done this before, please share your comments.