I just duplicate (CTRL/CMD-D) then drag it where I want it. You still have to drag it but at least you wont have to go searching for it first
Ok you have 110 folders in your Project and 300 comps and 1000 assets. Your goal is to copy or duplicate your Logo comp that lives in the Master folder, You also want to duplicate the background.mov file in the assets but you only wanted to use the first 3 seconds. You want to put those assets in the Version 3 folder.
Start by typing master in the search field. As soon as the Master folder shows up you can stop typing and add a comma and start typing logo until the logo comp shows up, then add another coma and start to type version until version 3 folder shows up. Add another comma and start typing background until the background.mov file shows up.
Now select the logo comp and press Ctrl/Cmnd + D to duplicate it. Rename it to anything you want. Drag it into the version 3 folder. Duplicate the background.mov file, rename that, then double click the duplicate to open the Footage Panel, move the CTI and use Alt/Option + [ to set a new in point and Alt/Option + ] to set a new out point, then close the Footage panel and drag your new shortBG.mov into the version 3 folder.
Now you decide you want to create a version 4 of you select the version 3 folder and press Ctrl/Cmnd + d to duplicate that, spin the folder open and rename the logo comp and the footage. You double click the footage to set a new in and out point for the footage. You decide you want to add some additional images so you press Ctrl/Cmnd + i to add more footage and place it in a new Additional Footage folder you put in the version 4 folder.
The last step is to delete the text in the search field and reveal your entire project.
EZ, Efficient, No muss no fuss and available since the search feature was introduced in AE and Premiere Pro.