In this work, we demonstrate universal multi-party poisoning attacks that adapt and apply to any multi-party learning process with arbitrary interaction pattern between the parties. More generally, we introduce and study $(k,p)$-poisoning attacks in which an adversary controls $k\in[m]$ of the parties, and for each corrupted party $P_i$, the adversary submits some poisoned data $\mathcal{T}'_i$ on behalf of $P_i$ that is still ``$(1-p)$-close'' to the correct data $\mathcal{T}_i$ (e.g., $1-p$ fraction of $\mathcal{T}'_i$ is still honestly generated). We prove that for any ``bad'' property $B$ of the final trained hypothesis $h$ (e.g., $h$ failing on a particular test example or having ``large'' risk) that has an arbitrarily small constant probability of happening without the attack, there always is a $(k,p)$-poisoning attack that increases the probability of $B$ from $\mu$ to by $\mu^{1-p \cdot k/m} = \mu + \Omega(p \cdot k/m)$. Our attack only uses clean labels, and it is online. More generally, we prove that for any bounded function $f(x_1,\dots,x_n) \in [0,1]$ defined over an $n$-step random process $\mathbf{X} = (x_1,\dots,x_n)$, an adversary who can override each of the $n$ blocks with even dependent probability $p$ can increase the expected output by at least $\Omega(p \cdot \mathrm{Var}[f(\mathbf{x})])$.