The Senate bill now moves to the House where an identical bill has already been passed. Each chamber is dealing with the other's (exact same) version of the legislation. Having already passed, each should move smoothly through the respective chambers.
The description of the bill on the Senate floor for a procedural vote on April 9 by its sponsor, Senator Will Kraus (R-8th) describes the reasoning behind the proposed March position on the calendar. A June date was considered but rejected as too late.2 And the original April proposal faced opposition from elections administrators because of logistical issues (e.g.: combining partisan and non-partisan races on one ballot, requiring two different machines). Not mentioned was the fact that this particular date in March will preserve the winner-take-all delegate allocation method Missouri Republicans have traditionally used. New RNC rules prohibit winner-take-all allocation of delegates before that point on the calendar.
The 25-7 vote split largely on party lines. Seven of the nine Democrats in the Senate opposed the measure. The majority Republicans were unified in support of moving the primary from February to March.
1 The specific move is from the first Tuesday after the first Monday in February to the first Tuesday after the second Monday in March. That would be a move from February 6 to March 15 for the 2016 cycle.
2 Presumably a June presidential primary would have been consolidated with primaries for state and local office as well. Separate legislation is seeking to shift the current August primary for state and local offices up to June.
Nebraska Bill on Binding Delegates Based on Primary or Caucus Results Signed into Law