Dam methyltransferase methylates large DNA molecules in a processive manner, with an effective range of up to 7 kb on either side of an initial methylation reaction.
My answer is taken from this paper:
Urig, S. et al. (2002) The Escherichia coli Dam DNA Methyltransferase Modifies DNA in a Highly Processive Reaction J Mol Biol 319 1085-1096
The authors report on a kinetic analysis of E. coli dam methylase. For anyone using this system as described by the OP, this paper is essential reading. Their conclusion is that the methylation process is highly processive, which is to say that a single dam-DNA interaction leads to multiple methylations before the enzyme dissociates from the DNA.
I'll restrict myself to describing just two of their experiments:
In this experiment they analysed the time course of methylation of an end-labelled 879-mer with 4 GATC sites, using DpnI digestion to detect methylation. In this experiment all DNA was found to be either fully methylated or unmethylated - no evidence for the presence of partially methylated DNA molecules was detected.
In this experiment they analysed the methylation of λ DNA (48.5 kb) and were able to detect partially methylated molecules. A kinetic analysis of the data results in the following conclusions (MTase; methyltransferase):
From these simulations, we [determine] a processivity of nav=3000. This means that after binding to the λ-DNA and starting the one-dimensional diffusion process, one MTase on average meets 3000 dam sites before it dissociates from the DNA. Since the enzyme performs a random walk, it will hit many sites more often than only once and the effective range of processivity corresponds to the square root of nav. Therefore, the dam MTase is able to methylate approximately 55 GATC sites on λ-DNA in a processive reaction in vitro. In this simulation, 4.6 associations of an MTase molecule to the same molecule of λ-DNA are required to obtain fully methylated DNA.
In a random DNA sequence GATC should occur on average every 44 = 256 residues. This suggests an approximate range of (55/2)*256 = 7 kb on each side of an initial methylation reaction.