When a diphthong ending with [u] or [i] is followed by a vowel, there is usually a transitory sound of some kind between them. After [u], it can be [v], [vv], or even [w], which does not otherwise appear in Finnish. After [i], it is [j] or [jj]. In some contexts, such transitory sounds appear after [u] or [i] even when it is not part of a diphthong.
The appearance, quality, and quantity of a transitory sound vary by the phonetic structure of a word, by language form, by speaker, and by situation. We can say that word kauan (a long time) may be pronounced as [kauan], [kauvan], [kauvvan], or [kauwan], but this describes just part of the variation. The length of the consonant may vary, too: it can be so short that the pronunciation can be between [kauan] and [kauvan], or it can be between [v] and [vv], or even longer than [vv].
Transitory sounds are written inconsistently, but according to specific rules. This means that in some words, v or j is written where a transitory sound may appear, in some words not, according to rules that depend on the type of word. To some extent, the variation in pronunciation may vary according to spelling, too; people may pronounce a transitory sound more clearly in a word like hauva than in a word like kauan.
The rules are:
Words where a transitory sound may appear between a non-diphthongal [i] and a vowel have special rules and special pronunciation. The transitory sound is never long, and when a [j] appears, it is often very short. However, j is written when it is part of a -jA suffix denoting an agent, e.g. kulkija (wonderer), a derivation of kulkea (to walk). This rule has been set to keep the suffix similarly spelled independent of the variation that the j is usually not pronounced after i. Due to the spelling, people may occasionally pronounce a full [j] in a word like lukijoissa, a form of the word lukija, to distinguish them from words like lukioissa (form of the word lukio).
In some dialects, a transitory sound appears e.g. after [u] in a word like kakkua so that it is pronounced [kakkuva]. When such dialects are written, the writing usually reflects pronunciation: kakkuva.