The vinyasas: The vinyasas connect the asanas. In it's classic form it is more or less a sun salutation. One speaks about full vinyasas and half vinyasas. Full vinyasa means that one starts from samsthiti and finishes the asana with samasthiti. Half vinyasa means that one returns always to a sitting pose to begin the next asana. I am a fan of full vinyasas.
To come to a sitting position from downward dog is not that easy. There are different possibilities: it can be done with legs crossed or with straight legs.
There exist different vinyasas. To go from tittibasana to bakasana and then to chaturanga dandasana is another one. It's only an example. Vinyasas connect the asanas, sometimes the classic connection must be modified.
In the Yoga Mala by P. Jois (the bible for the first series) the first series is described with full vinyasas. In most shalas, even in the shala in Mysore the yoginis practice half vinyasas.
Strength: The vinyasas build strength. Counter poses are done. Body control is learned. This is so good for daily life. I've no problems to lift my heavy suitcase, I open every jar. I don't have to ask for help anymore.
The vinyasas can be adapted to the ability of the student. There is much flexibility how to do it. At least they should be done between the asanas, between the sides is not necessary when the student is not able to do it. (I've heard so.)