Gareth Rees's answer is of course correct in pointing out that (a) the poem is unlikely to be Herrick's, and (b) the version in the question is truncated and with the punctuation altered. However, the last verse does make meaning in both versions.
First, the version the question cites:
Cupid is winged, and doth range
Her country; so my heart doth change.
But change the earth, or change the sky
Yet will I love her till I die!
Here, range is used transitively. Merriam-Webster says:
2 a : to rove over or through
b : to sail or pass along
So it means Cupid ranges over her country. In this case, her country is her body. This refers back to the previous stanza of this version:
Her gestures, motions and her smile
Her wit, her voice, my heart beguile;
Beguile my heart, I know not why
And yet I love her till I die!
Cupid roams over her gestures, her gait, her smile, etc., and each of those in turn beguiles the speaker's heart. So when the speaker watches her gestures, his heart loves her gestures; when he watches her smile, his heart loves her smile, etc. That is why his heart doth change. I disagree with Peter Shor's comment that the change from love to heart makes "total nonsense of the poem". (It made perfect sense to me when I read it.)
Next, the earlier version given in Gareth's answer:
Cupid is winged and doth range,
Her countrie so my love doth change,
But change she earth, or change she skie,
Yet will I love her till I die.
Here, range is used transitively:
1 a : to roam at large or freely
b : to move over an area so as to explore it
So the stanza could be interpreted to mean that the lady is moving to a different country. The speaker is saying that since the lady epitomizes winged Love, her moving is understandable. Yet no matter where she is, he will lover her forever.
That said, given how manuscripts circulated in the 16th and 17th C., with neither spelling nor punctuation standardized, I would hesitate to read too much into the presence or absence of a comma as indicating the correct meaning. Editorial choices such as punctuation or even spelling impose unitary meanings on texts, when the manuscript record or even early print records are much more ambiguous. For example, here is the 1623 Folio text of the first few lines of Hamlet's famous "To be or not to be" soliloquy:
To be, or not to be, that is the Question:
Whether ’tis Nobler in the mind to suffer
The Slings and Arrows of outragious Fortune,
Or to take Armes against a Sea of troubles,
And by opposing end them: to dye, to sleep
Here is how a modern editor, Harold Jenkins, normalizes these lines in the Arden edition, second series (1982):
To be, or not to be, that is the question:
Whether 'tis nobler in the mind to suffer
The slings and arrows of outrageous fortune,
Or to take arms against a sea of troubles
And by opposing end them. To die—to sleep,
No more; (III.i.56–61)
The insertion of a comma after sleep (in the last quoted line but one) guides the reader to a specific meaning: "To die is nothing more than to sleep." However, the Folio text is not as definite as that. A defensible interpretation of the Folio punctuation, which lacks the comma after sleep, is: "Dying means being unable to sleep any more." Perhaps Hamlet is saying that our life is something like a sleep, and what we think we experience is just a dream? Perhaps when we die, we wake up and so are free of the illusion that is life?
The point here is that the punctuation of early modern texts, being pretty sparse, is not a reliable guide to the intended meaning. The original sources can be open-ended. Later editors modernize the texts and insert punctuation based on their own interpretation of the meaning. William Empson (mentioned in a since-deleted answer to this question) points to the several possible readings of this last verse, which he punctuates as follows:
Cupid is winged and doth range;
Her country so my love doth change.
But change she earth, or change she sky,
Yet I will love her till I die.
He elaborates on the many ways that the last two lines could be interpreted:
"I will love her though she moves from this part of the earth to one out of my reach; I will love her though she goes to live under different skies; I will love her though she moves from this earth and sky to another planet; I will love her though she moves into a social or intellectual sphere where I cannot follow; I will love her though she alters the earth and sky I have got now, though she destroys the bubble of worship in which I am now living by showing herself unworthy to be its object; I will love her though, being yet worthy of it, by going away she changes my earth into desire and unrest, and my heaven into dispair; I will love her even if she has both power and will to upset both the orderly ideals of men in general (heaven) and the system of society in general (earth); she may alter the earth and sky she has now by abandoning her faith or in just punishment becoming outcast, and still I will love her; she may change my earth by killing me, but till it comes I will go on loving."
Empson, William. Seven Types of Ambiguity. 1930. 2nd ed. London: Chatto and Windus, 1947. pp. 48–49).
Ultimately, then, the version of the poem given in the question makes certain editorial choices in order to spotlight a particular meaning. The earlier text we have may or may not be the original. As Gareth says in his answer, it is not known whether the poem was written by Ford. Modern editions treat it as anonymous. So Ford's punctuation too may reflect his own choices. Hence, we cannot say for sure that Ford's version preserves the original meaning of the last stanza and the later, 19th C. version is simply wrong. The textual history of early modern poetry is too complicated for us to be confident about the real meaning of any given poem based simply on its earliest extant appearance in manuscript or print.