Although running entails moving, the Original Poster's sentence A), which involves running, does not entail sentence B), which involves moving. In other words sentence A being true does not guarantee that sentence B) is true. In fact the reverse holds.
Entailments and downward entailing environments
This question is far more interesting than it first appears. Under normal circumstances if A is a subset of B, A entails B. Running, of course, is a subset of the actions covered by the verb moving. For this reason the truth of sentence (1) below entails the truth of sentence (2):
- Bob ran.
- Bob moved.
So if we know that sentence (1) is true, we can be certain that sentence (2) is true. Of course, this will not work the other way round. Just knowing that Bob moved does not guarantee that he ran. Instead of running, Bob may have shambled, or danced, or writhed around on the floor. This relationship therefore does not run both ways. Sentence (1) entails sentence (2), but sentence (2) does not entail sentence (1).
Something interesting happens if we negate the sentence though:
- Bob didn't run.
- Bob didn't move.
Now the relationship seems to be reversed. Just because we know that (3) is true and that Bob did not run, it doesn't mean that we know that (4) is true. Bob may have been moving in all sorts of interesting ways without running. However, we can be absolutely certain that if Bob didn't move he didn't run! (This is because not moving describes a subset of the possible cases covered by not running.)
Environments which reverse the normal pattern of entailment, such as negative clauses, are called downwards entailing environments.
Now, conditional antecedents (the clauses in a conditional which follow the word if), are very similar in many ways to negative clauses. For example, they often take negative-polarity items. Consider the item at all in the following examples
- *You like Bertha at all. (ungrammatical)
- You don't like Bertha at all.
- If you like Bertha at all, come to her leaving party.
From the examples above we can see that at all doesn't work well in a positive sentence but can be licensed by a negative clause or by a conditional antecedent.
A second weird property of conditional antecedents is that, like negative sentences, they are downward entailing environments:
- If Bob ran, his shoe fell off.
- If Bob moved, his shoe fell off.
Here we see that the although running entails moving, sentence (5) being true does not entail sentence (6) being true. It might easily be the case that if Bob ran, his shoe fell of, but that if he scratched his head it did not. However, if for some inexplicable reason Bob's shoe did fall of whenever he moved, then it was also the case that Bob's shoe fell off if he ran.
Notice that this only happens in the antecedent. This is not true of conditional consequents (the consequent is the main clause of the conditional):
- If the bell rang, Bob ran.
- If the bell rang, Bob moved.
Here we see the normal running-->moving entailment. If we know that (7) is true, then we know, of course, that (8) is also true.
The Original Poster's example
A. If I run, I'll vomit.
B. If I move, I'll vomit.
As with the conditionals (5) and (6), the verbs run and move occur in a conditional antecedent. For this reason although running involves moving, the truth of (9) does not necessarily mean that (10) is true. In fact we see the reverse situation. Sentence (10) being true necessarily guarantees the truth of (9).