It is possible to migrate your .NET Framework ASP.NET MVC application to any version of .NET/.NET Core—including the latest version. Some parts of the process are more challenging than other parts, though, so you need to be prepared.
When developing a new application for the web, some precautions deserve attention, such as user experience, performance and security, among others. Check out this blog post from the ASP.NET Core Basics series for a collection of good practices to review when you start a new web project.
ASP.NET Core has a great code-sharing feature where you can easily share classes, functions and more between projects and thus save time in development. Check out this blog post on how to do this via NuGet packages and DLL files.