Neither sentence is incorrect, but the first one would benefit from using "would" instead of "will." It is a request, and there for "would" is more proper. "Will" is used for requests, but in more familiar instances (family/friends).
The second sentence could also use "would", but they instead chose to use "will". This gives the perception that if the person falls, someone must take over carry the flag. There is a condition, but the resulting action will definitely occur if the cause occurs.
"Would" and "will" are virtually interchangeable, but there are instances where one is more appropriate than the other. Here are some guidelines:
1.Would and will are both auxiliary modal verbs.
2.Will is used to talk about definite future actions.
3.Will is used to talk about: quick decisions, promises, offers and a likely prediction
4.Would is used to talk about: invitations, requests, asking permission, talking about preferences and making arrangements.
5.Both will and would can be used in conditional statements.
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