What is the known complexity of this game? (similar to PushPush-1)
There are no sliding blocks, so the game is in $P$

Informally: from the starting position just mark the cells reachable by the cursor. The level has a solution iif the final position becomes reachable. 

The algorithm is similar to the Dijkstra's algorithm for shortest path applied to the graph derived from the level.

For example:



For an $n \times n$ level the complexity is $O(n^4)$.

Let me know if you need further details.