When using MCF to fulfill orders, you'll need to make sure that the leadtime and shipping time are set on the non-Amazon channel to reflect the leadtime and shipping time of the fulfillment center you're using to avoid negative remarks on your non-Amazon marketplaces.
For example, if you have your leadtime set to 1 day on eBay, receive an order that is sent to FBA, which could have a 2 day leadtime, your eBay account could get dinged even though the order isn't late yet. All because the leadtime wasn't set correctly.
Most Amazon orders will ship within 2 days, but you will want to confirm that your Amazon account reflects that.
It is not a bad idea to "pad" your lead time and shipping time for your non-Amazon channels to allow for extra time. The worst that could happen is the order would make it to the buyer early!