When you make a parameter from a pre-made tool (e.g. Merge, Add Field, etc.) the parameter type is automatically set. So, if you make a model parameter out of the input box in the Merge tool, it automatically creates a parameter in your tool dialog that looks just like that in the Merge tool, that you can drag and drop into like usual, because it has chosen the parameter data type for you (check out the help page for each tool to see the parameter data types).
You can get into trouble creating your own parameters from model variables because you have to choose the data type yourself. Say you want to drag and drop some data from your table of contents and you choose Feature Class for the parameter data type - doesn't work! But, Feature Layer parameter data type does allow drag and drop, so try that.
A good way to get started figuring this out is to find a tool with the type of parameter action you're looking for, look up its help page, find the parameter data type, and use that.