How did the term "to favor" come to signify a limp?

The OED has this use of the verb favour dating back to the 16th century:

7. To deal gently with; to avoid overtasking (a limb); to ease, save, spare. Now colloq. (esp. in stable parlance) and dial.

1526 W. Bonde Pylgrimage of Perfection iii. sig. YYYiiiiv, Fauour thy body.

1590 R. Harvey Plaine Percevall sig. C4v, A Preacher..must haue his reader at his elbow, to fauor his voice.

1607 G. Markham Cavelarice ii. 42 When a horse doth stand but firme vpon..three feete..fauoring the other.

Earlier senses are: to regard with favour; look kindly upon; to treat kindly; to treat with partiality; to aid or support.

In that context, favor has the meaning “treat with care”; stated that way, the connection to other senses of the word is more obvious.

This usage it appears at least as early as 1826 in Robert Drury's Journal:

I walked on this seventh day; and though I favored my lame foot as much as I could, yet I rested but once all day. This way happened to be plain and easy.

However, this is the fourth edition of a book first published in 1729, so the usage may be considerably older, potentially predating Modern English. Favor first saw use as a verb in the mid-14th Century.