Fractal Chess

This game presents two boards at different scales, one with large squares corresponding to four small squares on the other one.
There's a similitude of play on both scale levels, hence the naming 'fractal chess'. The objective is to render some move more efficient at one scale, and see the consequences at smaller scale.

Setup

Two boards are needed. One of 8x8 squares, and one of 4x4 with double dimensions of squares. With simple tricks, one could transform a normal board into the latter.

Pieces

A normal set of men is used initially placed on the 8x8 board.

Rules

The rules of normal Chess apply to each board, exception made for promotion absent on the larger one.
In addition, pieces may switch to the other board empty squares (but not capture), although an attacked King must remain on his board.
The game is ended when the enemy King, on either board, is checkmated.

Notes

It is remarkable that on going from larger to smaller scale, two Bishop friends may finish on the same color (I don't know yet if this rule will be maintained, instead of keeping the normal rule of two opposite Bishop friends).
It seems that the King on the larger scale board is much more fragile than on the smaller one: so avoid it !
The moves seems also much more efficient on the larger board, in coming back later on the smaller one.