Token transfers and ownership on Matic versus Ethereum

I want to write a contract that mediates erc721 transfers. i.e. before a user transfers an erc721 to another user the contract checks against some requirements. If something like this is deployed on matic:

  1. can existing erc721s from ethereum mainchain be used or would i need to mint tokens specifically for use on Matic?
  2. if existing tokens from ethereum can be used; if an erc721 is deposited to a user on Matic can they use that token to interact with an existing game contract on ethereum main chain or would they need to withdraw from Matic first?

so for example: could a user deposit a cryptokitty to my contract; my contract then mediates transfers of that cryptokitty to new users on Matic. If a user on Matic receives the cryptokitty can they then use that cryptokitty to interact/play with the cryptokitty game/contract on ethereum or would the kitty need to be withdrawn from Matic first?

While moving an asset like cryptokitty to matic it would first be locked in ethereum mainnet. So I do not think you will be able to use the cryptokitty in ethereum mainnet when in matic unless you move it back to ethereum mainnet again.