We are working on an asset migration project from repos to AEM. The assets will be migrated to ACS however the existing site is available on AMS which will eventually be consuming these assets.
I am trying to understand if there is a possibility of supporting the existing Adobe managed services (AMS) site with this ACS DAM? If yes, how we can achieve it technically? Also, in future eventually the site will also move to ACS and if this mixed setup (if possible) can cause any potential issue at the time of site migration to ACS?
That's a common scenario and the integration between AEM Sites and AEM Assets is usually done through the Connected Assets feature . This documentation  has a matrix of supported setups including the architecture that you are looking for (AEM Assets as a Cloud Service and AEM Sites on Managed Services).