You can most definitely move the files from source to destination using the file system task.
Add the Foreach Loop Container to your workspace, and then the File System Task within it.
Create a variable in which to store your filenames.
Edit the Foreach loop first, select Collection from the left menu and then enter your Directory in the Folder space under Enumerator Configuration. For files, either keep the default if you are to move all files or add a mask if you wish to be selective.
Select the Variable Mappings tab from the left menu and then in the right menu, select the variable you defined to hold your filename from the first step and enter 0 in the Index field.
On the File System task, you will need to define a destination connection representing your destination directory, select the appropriate operation, and then change IsSourcePathVariable to true and select the SourceVariable from step 1.
The benefit of this method over the methods such as
COPY C:\YourSource*.TXT C:\YourDestination
is that it is much easier to automate this step among a series of tasks and workflows, using the created variables and components.