Hello Imre,
the "power transistor" on the MIDIbox is actually a voltage regulator of the type 7805, rated for a maximum current of 1A. When you draw 400mA and you have an input voltage of say 9 V, then it will dissipate 4V*400mA, which is just 1.6W. This doesn't seem much, but without a heatsink (which you installed now) it will get too hot. Fortunatley htese devices are quite well protected against overheating, current and even against short ciruiting them, so they are nearly impossible to destroy. But they do cut off the power when they get too hot, which is what you saw in your MIDIbox. So, no harm is done. If you actually want to draw more than 1A (by increasing the number of LEDs), you will need to use a stronger power supply.
Best regards
Reiner