I think your enthusiasm is great! But why do you think attracting people from other Stack Exchange sites is the best way to raise activity here? Don't get me wrong, I'm sure new users from other SE sites will improve activity, but if we are going to do more promotion, why not try and target the group with the most potential?
The 3 biggest SE sites, Stack Overflow, Super User and Ask Ubuntu, certainly have lots of users and I'm sure some of them will find this site interesting, but I'm not convinced programmers and system administrators are the best target groups for Sustainability SE. Most programmers I know are not interested in sustainability. I suspect that there are more potential new users for our site that have never even heard of Stack Overflow.
Number of visitors
Sustainability SE is now about 3.5 years old. As you can see in this post we went from about 100 visits per day to about 600 in 2013. This increased to 800 in 2014 and 1100 in 2015. I'm very curious to see how much visitors we have by the end of 2016.
You can also see in the statistics that the number of visits has its ups and downs. It looks like that June, July and August are the quietest months and November and January are the busiest. This means that right now is probably the most quiet time of the year here.
If I compare Sustainability with other beta SE sites I think overall we are doing okay. We are doing fine in terms of number of visits and users. Our weakest point probably is the number of questions asked, so instead of targeting new users we could also promote asking more questions by current users, for example by reviving the Biweekly topic challenge
Overlap with other SE sites
Like most SE sites we have overlap with other sites. Some questions can be on-topic on multiple sites. That's fine. We have questions here that would also fit on Gardening SE or Electronics SE, and I'm sure they have questions that would fit here.
Policy is to migrate questions only when they are off-topic on the site where they are asked, or perhaps if there is no answers after a while and you suspect people on the other site can provide an answer.
Attracting SE visitors
There are no special 'frameworks' as you call it, for questions that are migrated or for answering migrated questions. You could provide a bounty on interesting questions/answers yourself, but you can't do that too often as it will cost you reputation. Alternatively you could suggest methods to improve cross-site collaboration on meta.stackexchange.com, but you should probably check this related question first.
This being said, I think we can do several things to attract more visitors from other SE sites.
Link to relevant questions and answers. Occasionally I visit Gardening SE and Earth Science SE and sometimes I see a question that has been asked and answered here. When I see such a question I usually post a comment and link here. I'm not sure how effective this is to attract new visitors, but I suspect it helps a few people to become aware of our existence. I do think we should not do this too often because then it may become annoying.
(Ask to) add related feeds in chat rooms. A few years ago I arranged with Gardening SE to add a feed to their chatroom so that questions asked here and tagged with gardening will also be posted in their chat. We could try and arrange this for more related sites (if we have relevant tags).
Create our own community promotion ad. I've recently learned that at the start of the year, all mature SE sites run a community-add proposal on their meta site. Ads that adhere to the rules and receive a score of 6 or higher are then shown on that site for the entire year. Here's the meta post for 2016 on Chemistry SE for example. We could create an ad and try to run it at the start of 2017.
Add links in the wiki of closely related tags on other sites. We link to other SE sites in some tag wikis like the cooking tag for example. We could try and do something similar with tag wikis on other sites.
Ask more green-it questions. Since most StackExchange users have an IT background I think we can attract more users if we have more questions and answers that are related to sustainability and IT.
Attracting other people
I think that overall the best way to generate more traffic and attract new users is to ask good and interesting questions and provide good answers. I suspect most newcomers find this website via Google, so high-quality content is key.
You could also post links on related websites that direct to interesting content here, but be careful not to become a spammer. Personally I am sometimes active on Quora.com and a fair amount of my answers there have a link to questions and answers here.