[UmpireErnie]

Originally Posted by nopachunts

Correct batting order: B1, B2, B3, B4
Batting order for this scenario: B1, B3, B2, B4.

At the start of the batting order: B1 batted and reached base and a pitch was made to B3, B3 is now the proper batter without regard to B2 not batting.
No. B2 is the proper batter while improper batter B3 is at bat. Proper batter B2 could replace improper B3 at any time prior to improper B3 completing her improper turn at bat. But that didn't happen here instead improper B3 got on base.

If B3 gets on base, B4 would be the proper batter.
Not until a pitch is thrown to a batter following improper B3s time at bat. This is the defense's chance to appeal BOO: from the time improper B3 gets on base till the next pitch is thrown.

If B2 stepped into the box, and the defense appeals that B2 is the incorrect batter before a pitch is made, B2 goes back to the dugout and B4 comes to the box, no out assessed.
Improper B3 has just reached base, and B2 comes to bat. Defense appeals BOO before a pitch to B2. The appeal is honored. The batter called out is the batter who was supposed to bat when improper B3 batted, in this case it was B2. B3s improper turn is negated, other runners who advanced are returned. Since B2 was called out for missing her turn the next batter in order is B3

In your scenario, if a pitch is made to B2, B2 is now the proper batter. Not yet. B2 came to bat after improper B3 got on base. Throwing a pitch to B2 after B3 gets on makes improper B3s turn legal and the correct batter would be B4 as she follow B3. But B2 continues to bat..

B2 gets on base. The proper batter after B2 reached base is now B4 as B3 is already on base and that spot in the batting order is skipped over without penalty.
B4 was proper batter during B2s improper turn. If defense appeals before a pitch is made following B2 getting on base then B2s turn is negated and B4 is called out. Since B5 follows B4, B5 is now proper batter.

But if a pitch is thrown after improper B2 gets on and B3 is currently on base then yes skip over B3 and B4 is proper batter.

Dakota had is right.