What are approach control's expectations regarding weather information for flights to satellite airports?
I fly out of a controlled satellite class D (KLOU, Bowman Field) below a class C (KSDF, Louisville International) so I'm fairly familiar with this scenario, at least in the way that it's handled there. The short story - in my experience - is that Approach only cares about you having the numbers or ATIS for the airport where you'll actually land.

First the theory. The ATC orders say:


  7−1−3. APPROACH CONTROL SERVICE FOR VFR ARRIVING AIRCRAFT  
  
  Issue the following where procedures have been  established for
  arriving VFR aircraft to contact approach control for landing
  information:  
  
  a. Wind, runway, and altimeter setting at the airport of intended
  landing. This information may be omitted if contained in the ATIS
  broadcast and the pilot states the appropriate ATIS code or if the
  pilot uses the phrase, “have numbers.”


Note the "airport of intended landing", they aren't going to pester you to get the ATIS for somewhere that you aren't going to.

Second, the practice. If I'm inbound to KLOU on flight following (don't leave home without it!) then my first contact with Louisville Approach usually goes like this:


  Me: Louisville Approach, N12345 level at 6500
  Appch: N12345, roger, Louisville Altimeter is 2992, report when you have the weather at Bowman, information Kilo is current
  Me: N12345 has Kilo at Bowman
  Appch: N12345, roger [and further instructions as needed]


As a controlled field, Bowman has ATIS, not "the numbers" but I've had more or less the same exchange at other places. 

I think that Ron's comment about information is useful here: when I'm inbound I usually have both the KSDF and KLOU ATIS anyway, just in case. Also notice how the KSDF controller helps me: the KLOU ATIS could have changed from Kilo to Lima between the time I got it and the time I called Approach. By telling me that Kilo is current, the controller is effectively telling me if I have to go and get the ATIS again or not.

As for just passing through class C, the very few times I've done that ATC has simply given me the local altimeter setting and that's it. I'm not landing, so they don't seem to care if I have the ATIS or not.