If you currently have a fixed tenancy agreement, then that remains as it is until the end of the fixed period, they can do nothing about that.
Once the fixed period ends, the tenancy becomes a periodic tenancy, that is a rolling month by month tenancy.
Ghis will end either by you terminating or a new fixed tenancy being created.
If the new landlord does not want a periodic tenancy but wants security of a fixed period tenancy, then yes they can do that. They can with a new fixed tenancy include a deposit, that is very normal, they can also include a new payment date.
You do not have to agree with this. If you do not, then the landlord has several options.
Either leave things as they are, or serve you notice to terminate the tenancy and for you to leave the property.
Your option is to accept the new fixed tenancy, or to say as you are and see what the landlord does, or leave and move to a new property.