The weather on the Via Francigena will depend on the season and the region you choose to travel. Some parts of Italy can reach very high temperatures during Summer, but from Autumn to Spring you will find colder days, even very cold ones.
During late Autumn and Winter some hotels might close and flights and ferries operate less frequently than in the warmer months. Daylight hours are also reduced, so you will need to adjust and manage better the daily distance you want to cover. The weather on the Via Francigena starts to get milder in Spring. Although nights and early mornings can still be pretty cold and rainy, especially on the coast and the mountains in general.
The rainiest months are definitely in Autumn but Spring can also be pretty rainy. In the Rhone-Alpes region in France, it is not unusual to get some rainy days in the Summer also.
The warmest months enjoy much longer days in terms of light hours, rain is infrequent and temperatures are higher. The weather is mainly dry and warm most of the way. Keep in mind that some days can be very hot and sun protection is vital. Also during the night, you might suffer from very warm temperatures.
Bear in mind some sections might be too hot for walking in July and August, for example, the Via Francigena across Aosta and Vercelli.
Some sections are not recommended during Winter season: from October/November to February/March, as there might be snow in high altitude and it could be very difficult to walk. This is the case in sections crossing mountains, such as the Via Francigena across the Alps in Switzerland. During the coldest months, this section is usually completely covered by snow and can get quite foggy. For this reason, it would be very difficult and unpleasant for you to walk and for us to transfer your luggage.
The landscape, climate and weather on the Via Francigena is very varied, not just from one country to another, but also from one city to the next, even between places not that far from each other.