I know that $(k,n)$ can increased to $(k+t,n-s)$ one, by generating a random polynomial $p(x)$ of degree $k+t$ with constant term $0$, and then ordering each agent $a$ to add $p(a_x)$ (where $a_x$ is an agent's public x-coordinate) to their share (if $k$ old shares are compromised, the secret is compromised, but we are assuming enough old shares are destroyed to prevent this) (we get $n-s$ by either ordering $s$ agents them to destroy their share, or, if it is already compromised, not sending an update to that agent.)
Also, I know a $(n,n)$ XORing secret sharing scheme can be increased to $(n+t,n+t)$, by generating $n+t$ bit strings that XOR to $0$, and sending the first $n$ to each agent to XOR with their current share (and send the rest to $t$ new agents). This in particular means there exists a secret sharing scheme $(k,n)$ can be updated to $(k+t, n+t)$ via this.
Can a $(k,n)$ shamir secret sharing scheme can be updated to a $(k+t, n+t)$, without any agents having to reveal their share?
Note: Remember, we are assuming that enough old keys can be destroyed to prevent the secret from being reconstructed from old keys alone (which is why this doesn't apply.)