How Do I Make A Beam Cross a Barline Into the Next System?

Musescore currently cannot do this by default. However, there is a work around.

Add an extra eighth note rest at the end of the measure before the system break, and an extra rest at the start of the measure after the break. You can do this by selecting a bar and doing Right Click > Bar Properties and changing the actual length (to 7/8 in this case).
Select the notes and the rest and select beam together (in the palette or Properties). This beams the note with the rest:

Then, make the rests invisible. Select one, and reduce its leading space (changes the length of the beam). Then, offset it and use the barline's leading space options to move the last notes back into their original position.

The same steps work on the following bar, but you use the note's leading space to control beam length and the rest's leading space and offset to move them left.

Messing around with these options you should be able to get something that looks nice, but at the sacrifice of playback :) Hope this helps!!!