Here is a weather site that has relatively accurate predictions and climate history:
And the climate chart on the Wikipedia pages should give you a good idea what to expect (eg for Cusco):
You should be fine as long as you use proper precautions and prudence. Keep an eye on your belongings and don't go wandering around on your own late at night etc etc.
I don't know anything about the trip between Peru and Ecuador by land but I should imagine its pretty easy. Just don't underestimate the distances! E.g. only the trip between Lima and Cusco takes over 20 hours by bus.
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