There is a reason to move and not convert code to .NET Core. On the one hand, .NET Core allows web applications to scale themselves (resize) to run on Linux, having more .NET Framework will make .NET Core more useful. But on the other hand, developers who want to use Microsoft technologies like WinForms, Windows Presentation Foundation or ASP.NET will be forced to use the .NET Framework.
The transfer should be done step by step. For example, putting an ASP.NET MVC application developed on Windows Server to ASP.Net Core on Linux via Azure.
Microsoft recommends uploading to ASP.Net Core while still targeting the .NET Framework. You can then switch to .NET Coreduf that is still on Windows. Finally, move to Linux and Azure. (ASP.Net .NET Core extension for web development)
Note that the order of steps can vary greatly. Microsoft also provides instructions for moving https://docs.microsoft.com/en-us/dotnet/core/porting/ including how to identify dependent assets on third parties and using API Portability Analyzer tool. https://github.com/Microsoft/dotnet-apiport/
See also: Instructions for installing the .NET Framework 3.5 100% successful on Windows 7