My wife often sends short videos to me in text messages of our daughter during the day. They come through fine in iMessage. The alternative I suppose is to use Whatsapp.
Shared PhotoStream - create one for pics/vids of your daughter. Your wife can add them to it and you can add whomever you'd like to the Stream.
I'm really looking forward to this feature when my wife and I have our first child. Being able to share big moments with both our families across the country in a matter of moments in a relatively private manner (as opposed to FB) is really cool to me.
Plus I travel - so she'd be able to share stuff I might miss.
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I know people love to trash it but Facebook Messenger works great for sending video and pics. It really has gotten a lot of features in the last year or two (being able to send videos being one of them).
Unlike needing the other party to have Whatsapp, etc, most everyone has facebook. Doesn't matter if they are on Android or iOS.
Michael
Unless you're trying to send something to my wife who refuses to download the additional FB Messenger app as a protest or something.....lol. She has like a dozen unread messages on there and says if the person needs to get a hold of her they can text/call her or send an email haha....
But I agree.