The structure of $C_i$ gives a hope that $C_i$ can be described as a low-rank update. If that is the case (size of the square block is much smaller than the size of the matrix $B$), one can apply classical methods for low-rank updates based on Sherman-Morrisson-Woodbury identity.
While updating the LU factorization is not the easiest task (the required pivoting might be sensitive to the performed modification) it is certainly possible. To begin with, you might want to read the classic low-rank update section of QR in Golub, Van Loan (Section 12.6 in 2nd edition). Then, proceed with the intricacies of updating the $PA=LU$ factorization.
Note, since $B$ is symmetric, one can take advantage of Cholesky factorization whenever $C_i$ is also symmetric.
However, if $C_i$ is not (enough) low-rank, I don't think there is much hope on any savings here.