If you want to redeem miles for flight awards there is no need to move your miles. You can use LH miles to redeem for any available award flight on any Star Alliance carrier, such as on AC.

I have no idea if M&M is better for miles redemption. As far as miles earning goes, that depends.

M&M has much better bonuses for paid J and F travel, but has more economy fares that earn 50% miles.

With AC all international flights earn 100% miles, but the J/C/Z COS bonus is not as generous. The lowest domestic economy fares earn 25% non-status miles, but it has more 100% miles earning fares than LH does.

If you are interested in earning bonus miles from flying that are available to frequent flyers, AC and LH do give 25% bonus miles to its own FFers for flying on the other airline, but you might get better bonuses for being a FFer flying on the airline with which you have the status.

For earning miles not from flying, well that just depends on where you live (or at least have an address). It is easier to earn AP miles in Canada and I assume easier to earn M&M miles in LH's empire of airlines.