They have to qualify for entry into the stream to receive any of the assets in the stream. People will automatically flow from one email to the next within the stream depending on the cadence you set.
Why you're adding the default program to the nurture stream? Even if the program is outside of the engagement program, Marketo won't send lead through the Program in the Stream more than once. You could use a list filter to identify the leads that are members of the program to identify who enters the stream, if there's a reason a lead should skip a stream then you can identify that in your transition rule between streams so they will receive something in each cast.
So here is my example:
Say I have 1000 people in my stream and I have 6 emails I want to drip to them with a cadence of 1 email per week. Then I have a market manager request that for a particular subset of those 1000 people, lets say 200, they get a 7th email in between emails 2 and 3. I can create that 7th email in a default program and have the smart campaign within it targeting those 200 people. Then I can drag that program into the stream in-between emails 2 and 3. Then as those 1000 people move through the stream, when it comes to casting out the default program, Marketo will see who qualifies for the program based off the filter criteria (in this example 200 people) and send out that 7th email. And then as more people get added to the stream, they follow this path and if they qualify for the same filters that separated out the initial 200, they too would get the 7th email.
What I am unsure of, is what happens to the remaining 800 that didn't qualify. Do they get moved on and sent email #3? or do they get nothing for that week?