This is a typical handover from the male to the female, the bird at the bottom of the picture being the male. The male normally finds the nesting place and invites the female in to give it the once over. If she is happy with it, than that is that!
Since I last posted, bluetits have occasionally gone into each of our three boxes, but nothing dramatic like another fight has occurred. Here for the first time we can see a pair of bluetits in Box1