In my day that was the battalion IntO & OpsO's job for red and blue respectively. Arty would contribute data but the only ones that touched the maps were the Bns Int and Ops folks. Same, same at Bde level.
Maintenance of the Common Operating Picture (COP)/CURSIT is the responsibility of Operations. At Formation level the G3 branch will be responsible for the integration of the various aspects of the CURSIT. Other Branches will have staff responsible for maintaining their slice of the COP (Int, Fires, Engr etc). The digital COP is the record - a map table/wall map with pins/unit icons will be the backup. I find that planning is usually done on either a paper map or a PowerPoint map chip and then "translated" into Battleview once the plan is finalized. The CURSIT map table is often used for really short term planning where it can be easier to get a few people around the table and trash something out. We had a digital map table at 1st Cdn Div that could be used for that as well.
The LCSS automatically feeds locational data upwards from enabled vehicles in the platoons/troops etc. At the present moment the Battle Group is the digital divide. The BG CP pushes info using voice to the sub-units who are maintaining maps (in a CP or in their turret). Battlegroup, Bde and Div can share graphics digitally, but from BG down to the sub-units its the old way. Having said that, I have been part of some experiments where it was pushed down to Pl/Tp level.