two schools of thought, either bed it, make the barrel solid to the stock or leave as is "free floated", some work better either way,,,,the proof is in the firing and the accuracy you attain.
If the receiver moves with the barrel, no.... If the barrel moves IN the receiver, the v-block (retains the barrel to the receiver) is probably loose and needs to be torqued.
The OEM stock should have two small swells in the front of the barrel channel to put upward pressure on the barrel when the action is drawn down by the action screw.
Try this. Loosed the action screw one, maybe two turns. Now place your thumb/finger against the action at the front as you tighten the action screw back to where it was when you started out. You should feel the action pull down as the screw is tightened. There should be about a 0.010" +/- of movement.
Yes, you can move the barrel if you push on it, this is normal.
Bead the stock as mentioned... then you can use a big nylon wire tie... NAPA sells them and they are black.... for the barrel band...
Most target shooters say the barrel needs to be free floating on a 10/22 and they are probably right... but I like a tight fit between the barrel and the fore stock... just feels better to me.