I took a look at actual scientific studies that examined oral heath of dogs. Unfortunately most studies are paywalled, meaning only the Abstract is publicly available but you need to pay money to read the entire paper. If you're interested in reading the papers anyway there is a free alternative that I will not promote here because it's a legal grey area (rather dark grey TBH).
Summary
It's true that a high fiber diet or special dental chews can reduce plaque in dogs, but the most effective treatment is daily tooth brushing (and probably the cheapest).
Since dental chewing treats are shown to be effective, it's plausible that high fiber dental food that is fed as a treat is effective as well, but no study actually tested this idea.
Since the special foods increase oral hygiene by brushing the existing biofilm from the teeth, there should be no difference in feeding it before or after the regular dry food. However, I do expect a better result if the special food is fed after feeding wet food, since wet food may leave more residue in the mouth after feeding.
Studies
PREVENTION OF PERIODONTAL DISEASE by Gregg A. DuPont, DVM
The study explains why the only way to avoid plaque is to brush or wipe away the biofilm that forms on teeth. It states:
Tooth brushing remains the proven most effective means of removing
plaque and, therefore, preventing gingivitis. Tooth brushing has been
shown to remove subgingival plaque down to approximately 1 mm from the
gingival margin. Periodontal health has been related to routine brushing. Our
clients should also brush their pets' teeth as a daily routine.
You need to get your dog used to daily brushing. Although it's best to start at puppy age, even old dogs can get used to teeth brushing. Although there are pet flavored tooth pastes available, you don't even need to use any.
Wet food doesn't have any abrasive effect and contributes to much more plaque than dry kibble or dental food. Certain rubber or raw hide chewing toys can have the same effect, if the dog chews on them long enough. A natural diet of raw meat can also decrease plaque, but certain breeds are more predisposed than others and chewing food doesn't offer them the same beneficial effect.
Effect of a New Dental Hygiene Chew on Periodontal Health in Dogs by Cecilia Gorrel, BSc (Dent), VetMB, DDS; Janice Warrick, RLATg; Tiffany L. Bierer, PhD
The final result of this study is less interesting than the observation during the preparation and test phase. All dogs received daily tooth brushing in preparation for the study. During the actual test phase they received no tooth brushing and only the test group had 2 dental chews daily in addition to the same dry food the control group had. (Emphasis added by me)
As soon as toothbrushing ceased (start of test phase), plaque accumulation and gingival inflammation increased. The amount of plaque deposited after adding the dental hygiene chew to the dry diet was significantly less after 4 weeks, compared to feeding the dry diet alone.
Effects of a Dental Food on Plaque Accumulation and
Gingival Health in Dog by Ellen l. Logan, DVM, PhD; Oliver Finney, PhD; John J. Hefferren, PhD
This study examined 40 dogs (34 mongrels and 6 beagles) for a period of 6 months. The test group was fed a special oral health diet and the control group was fed a regular dry kibble.
The dogs of the test group had 40% less plaque than the dogs of the control group and 36 - 50% less gingivitis. The study also shows that the beneficial effect of a dental cleaning lasts only 4 weeks.
Oral Health Benefits of a Daily Dental Chew in Dogs by Bradley W. Quest, DVM
They observed 60 beagles (male & female) over a period of 28 days. All dogs had their teeth cleaned by a vet at day 0 of the study. All got the same dry kibble and the dogs of the test group got a single dental chew daily in addition to their food.
The study found that the dogs who got the dental chew produced 32% less plaque during the 28 days period and had 80% less gingivitis. Be aware that the authors are heavily biased towards finding positive results because they belong to the same company that produced the dental chews.