After a match has taken place, you can enter the results through the Competition Builder and advance the winner onto the next round automatically.
To do this, switch the Competition Builder from "Edit mode" to "Results mode". This is done in the top right hand corner of the Competition Builder screen, next to "Settings".
Once in "Results mode", a new stopwatch icon should become available next to each match. Click this to open the results screen.
From here you can enter the score of each team (or select a forfeit if applicable), and choose the "Progressor". Select "Enter & Progress" to finalise the match.
If there is a round following the current round and it is set to "Auto Map Progression from Preceding Round", this progressor will automatically move into the next round. 
Drag and Drop
You can quickly remove a school, competitor (or any other entity such as a bye) by simply selecting the trash can on the right hand side.
A free slot will display the text "Drag Entity".
To select an entity to drag into this slot, select the dropdown menu "Match Candidates".
You can select "Display All" if you wish to see all applications available to you, including those schools that are already allocated to matches.
You will be able to pick up BYE, BLANK, all the schools and also the placeholder match winners and place them into any empty slot.
Please note: You will not be able to drag entities in blank spots (you will need to trash the blank first).