Why is $O(nk)$ an upper bound for the $k$-gossip problem?

I am studying the $$k$$-gossip problem on dynamic graphs against an adaptive adversary. Essentially, we are given a set of tokens $$\mathcal{T}$$ which are distributed amongst the nodes such that each token is distributed to at least one node. Importantly, the nodes do not know the value of $$k$$. The problem is solved when all nodes know all $$k$$ tokens.

The paper Information Spreading in Dynamic Networks write in their abstract:

Our main result is an $$\Omega(nk/\log{n})$$ lower bound on the number of rounds needed for any deterministic token-forwarding algorithm to solve $$k$$-gossip. This resolves an open problem raised in , improving their lower bound of $$\Omega(n\log{k})$$, and matching their upper bound of $$O(nk)$$ to within a logarithmic factor.

The paper they mention for the $$O(nk)$$ upper bound gives a protocol for $$n$$-gossip but not for $$(k-gossip. Is there an easy way to see the $$O(nk)$$ upper bound for $$k$$-gossip for dynamic graphs?

We know that $$1$$-gossip can be solved in $$n-1$$ rounds, where $$n$$ is the size of the network if the nodes know $$n$$. So could we maybe do $$1$$-gossip $$k$$ times. The only issue is that the nodes do not know $$n$$ or $$k$$, so we can't use this approach.