Will existing MVC projects/pages continue running into .NET Core, so my team can slowly transition to .NET Core?

Yes, it is perfectly OK to use your existing MVC pages with .NET Core. In this case, the .NET Core renderer will act as a proxy and serve the MVC pages to the client, so you can have MVC and .NET Core pages within the same project trouble-free. This approach is especially helpful when trying to migrate existing projects to .NET Core. 

As for mixing different types of widgets, you cannot place your existing MVC or WebForms widgets on a.NET Core page or template. Only .NET Core widgets can exist on .NET Core pages. Keep in mind that the paradigms of MVC and ViewComponents are similar enough for some code to be reused. 

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