Julf is right on.

Use Ohms law to get to the current by measuring the voltage over the resistor instead.

5.7mA and 1kOhm should mean you have a voltage of 5.7V across the resistor - and on your turn on terminal/lead! That is quite a bit - if it actually is 5.7mA...
Considering you only measured about a volt on it before, you might be an order, or two, of magnitude off. (with the real current even lower, like 0.57mA or something like that...)

/Michael

What you need to do is put the meter in series with the circuit you want to measure, not in parallel.

I did. I clamped the end of the wire to the mm and connected the other pole of the mm to ground.
IOW: reomote wire of the majestic amp is connected to an empty lead and a lead with the 1kOhm resistor. I measured the voltage on the empty lead with and without the resistor wire being connected to ground. After that, I switched the plug on the mm for current measurement and connected one end to ground and the other to the wire with the resistor. That's when I got the weird results. I'll do it again right now and see if I can make sense of it.

Use Ohms law to get to the current by measuring the voltage over the resistor instead.

Great idea! I should've known that myself!
/me hits myself repeatedly in the forehead
I now found out something even more interesting: the voltage on the majestic remote-turn-on lead actually goes to 11.5V immediately if I take the empeg out. It goes back to slowly increase, if I put it in and stays 0 if I ground the 1kOhm wire. All the time, the ignition is off and the empeg, therefore, is also off. The turn-on wire is not connected to either the empeg, nor the other amp. Thus, the only connection between the empeg and the majestic amp are the RCA cables. What does it mean that the voltage disappears, if the empeg is plugged into the sled, vs. it being out of the sled? The majestic amp is grounded to chassis with a very short wire (about 5 inches, 10-15cm). The "front" and "rear" RCAs go into a splitter at the Memphis amp and one end of the "Y" goes into that amp and the other into the majestic amp. I find that all very confusing...