1.The time between innings is not a suspension of your responsibilities
2.You don't want to look sloppy and unprofessionally during that space of time
3.You want to be alert on whatever happens out there: substitutions, not enough players on the field, players who take ages to get ready for the next inning, etc.
4.You'll have to count warmup-pitches
5.You don't want to miss the moment when players are ready to resume play
However, some other points require a closer look.
For example:
Another fielder picks up the ball, and throws a warmup-pitch,"just for fun"
One umpire wrote: "One thing you particularly want to observe is the pitching mound. During the inning change, occasionally another fielder, say "Jimmy," the Senators' left fielder, is jogging past the mound and he will see the ball laying there where the other team's catcher tossed it when the last half inning ended. Jimmy has been wanting to pitch but the coach hasn't let him try yet. So just for fun, Jimmy picks up the ball, steps on the rubber and tells "Charlie," the catcher to catch one pitch for him. Charlie's got nothing to do since, "Elvis," the real pitcher, still hasn't come out onto the field. So Charlie squats behind home plate and Jimmy lets one fly. After Jimmy's "pitch," he turns and resumes his jog out to left field."
"Jimmy has just legally become the Senators' next pitcher. Rule 3.08 (a) (1). Not only does Jimmy have to face at least one batter, but Elvis can no longer return to the position of pitcher in this game."
I beg to differ! Rule 3.08(a)(1) covers substitutions only. A substitution is the replacement of one player who has been in the line-up (or pitching, in case of DH) for another player, who has not been listed in the line-up so far. This rule doesn't cover change of positions. As far as I know, there is even no requirement for managers to notify the umpire in case of a change of positions between fielders.
Therefore, Jimmy is not required to pitch from now on, and Elvis can resume his "job".
The above note doesn't imply that it's a good idea to have fielders juggle or pitch the ball just for fun. Umpires should tell them to get to their positions, instead. Why? Suppose the ball is pitched by Jimmy, and sails way over the catcher's head.... it'll take a while before he gets the ball back, which delays the game, and looks bad.
The right spot for a sip of water
Some umps advise us to stay out of the dugout. If you really want to sit down and have a sip of water, you should do so in the dugout of the team that just comes off the field, because they just retired a batter or runner for the third out, and should be happy.
Not a bad thought, but there are some good reasons for taking the other dugout, too. The main reason I would like to bring up, is, that the other team will now be on the field! At the lower levels, the dugout at that side of the field will be almost empty, and chances are, that you can find a spot to sit without having to dodge the player who just made that terrific dive in the mud. Another reason may be, that you can easily have a quick look at the on deck area nearby, to see if there's still some equipment lying around.
The base umpire's position between innings
Some umpires totally overlook the base umpire's tasks. In my opinion, they should be:
1.Watch the players come on and off the field. Especially after a half-inning when collisions occurred, or players got involved in some heated arguments, the umpiring crew must make sure that players don't "coincidentally meet" while coming from or going to their positions.
2.Make sure the defense is placing nine players on the field, and make sure they're all wearing a proper uniform
3.Make sure that balls are thrown back to the dugout before signalling to the plate man that everyone is ready to resume play.
The Umpire Development Program advises base umpires to assume a position on the edge of the grass between first and second (second and third) base, at the outside of the infield. However, considering the tasks mentioned above, I don't think this is the best spot to be in. Instead, you might prefer to jog some 80 feet along your foul line towards the foul pole and stand there between innings. Make sure, though, that you're not in the way of outfielders warming up.
From there, you have an excellent view of the whole field, and you can see each and every player. Another important advantage is, that you're at quite a distance from the dugouts and coach boxes. No-one will try and approach you in order to discuss calls you made in the previous half-inning. And that, of course, will enable you to relax and get ready for yet another interesting inning of baseball.