  We study the convergence in probability in the non-standard $M_1$ Skorokhod
topology of the Hilbert valued stochastic convolution integrals of the type
$\int_0^t F_\gamma(t-s)\,d L(s)$ to a process $\int_0^t F(t-s)\, d L(s)$ driven
by a L\'evy process $L$. In Banach spaces we introduce strong, weak and product
modes of $M_1$-convergence, prove a criterion for the $M_1$-convergence in
probability of stochastically continuous c\`adl\`ag processes in terms of the
convergence in probability of the finite dimensional marginals and a good
behaviour of the corresponding oscillation functions, and establish criteria
for the convergence in probability of L\'evy driven stochastic convolutions.
The theory is applied to the infinitely dimensional integrated
Ornstein--Uhlenbeck processes with diagonalisable generators.
